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Theorem infxpenlem 8172
Description: Lemma for infxpen 8173. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
leweon.1  |-  L  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) }
r0weon.1  |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }
infxpen.1  |-  Q  =  ( R  i^i  (
( a  X.  a
)  X.  ( a  X.  a ) ) )
infxpen.2  |-  ( ph  <->  ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) ) )
infxpen.3  |-  M  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )
infxpen.4  |-  J  = OrdIso
( Q ,  ( a  X.  a ) )
Assertion
Ref Expression
infxpenlem  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
Distinct variable groups:    A, a    w, J    z, w, L   
z, m, M    ph, w, z    z, Q    m, a, w, x, y, z
Allowed substitution hints:    ph( x, y, m, a)    A( x, y, z, w, m)    Q( x, y, w, m, a)    R( x, y, z, w, m, a)    J( x, y, z, m, a)    L( x, y, m, a)    M( x, y, w, a)

Proof of Theorem infxpenlem
StepHypRef Expression
1 sseq2 3373 . . . 4  |-  ( a  =  m  ->  ( om  C_  a  <->  om  C_  m
) )
2 xpeq12 4854 . . . . . 6  |-  ( ( a  =  m  /\  a  =  m )  ->  ( a  X.  a
)  =  ( m  X.  m ) )
32anidms 645 . . . . 5  |-  ( a  =  m  ->  (
a  X.  a )  =  ( m  X.  m ) )
4 id 22 . . . . 5  |-  ( a  =  m  ->  a  =  m )
53, 4breq12d 4300 . . . 4  |-  ( a  =  m  ->  (
( a  X.  a
)  ~~  a  <->  ( m  X.  m )  ~~  m
) )
61, 5imbi12d 320 . . 3  |-  ( a  =  m  ->  (
( om  C_  a  ->  ( a  X.  a
)  ~~  a )  <->  ( om  C_  m  ->  ( m  X.  m ) 
~~  m ) ) )
7 sseq2 3373 . . . 4  |-  ( a  =  A  ->  ( om  C_  a  <->  om  C_  A
) )
8 xpeq12 4854 . . . . . 6  |-  ( ( a  =  A  /\  a  =  A )  ->  ( a  X.  a
)  =  ( A  X.  A ) )
98anidms 645 . . . . 5  |-  ( a  =  A  ->  (
a  X.  a )  =  ( A  X.  A ) )
10 id 22 . . . . 5  |-  ( a  =  A  ->  a  =  A )
119, 10breq12d 4300 . . . 4  |-  ( a  =  A  ->  (
( a  X.  a
)  ~~  a  <->  ( A  X.  A )  ~~  A
) )
127, 11imbi12d 320 . . 3  |-  ( a  =  A  ->  (
( om  C_  a  ->  ( a  X.  a
)  ~~  a )  <->  ( om  C_  A  ->  ( A  X.  A ) 
~~  A ) ) )
13 infxpen.2 . . . . . . . 8  |-  ( ph  <->  ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) ) )
14 vex 2970 . . . . . . . . . . . . 13  |-  a  e. 
_V
1514, 14xpex 6503 . . . . . . . . . . . 12  |-  ( a  X.  a )  e. 
_V
16 simpll 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  ->  a  e.  On )
1713, 16sylbi 195 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  a  e.  On )
18 onss 6397 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  On  ->  a  C_  On )
1917, 18syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  a  C_  On )
20 xpss12 4940 . . . . . . . . . . . . . . . 16  |-  ( ( a  C_  On  /\  a  C_  On )  ->  (
a  X.  a ) 
C_  ( On  X.  On ) )
2119, 19, 20syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( a  X.  a
)  C_  ( On  X.  On ) )
22 leweon.1 . . . . . . . . . . . . . . . . 17  |-  L  =  { <. x ,  y
>.  |  ( (
x  e.  ( On 
X.  On )  /\  y  e.  ( On  X.  On ) )  /\  ( ( 1st `  x
)  e.  ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  e.  ( 2nd `  y
) ) ) ) }
23 r0weon.1 . . . . . . . . . . . . . . . . 17  |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }
2422, 23r0weon 8171 . . . . . . . . . . . . . . . 16  |-  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) )
2524simpli 458 . . . . . . . . . . . . . . 15  |-  R  We  ( On  X.  On )
26 wess 4702 . . . . . . . . . . . . . . 15  |-  ( ( a  X.  a ) 
C_  ( On  X.  On )  ->  ( R  We  ( On  X.  On )  ->  R  We  ( a  X.  a
) ) )
2721, 25, 26mpisyl 18 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  We  ( a  X.  a ) )
28 weinxp 4901 . . . . . . . . . . . . . 14  |-  ( R  We  ( a  X.  a )  <->  ( R  i^i  ( ( a  X.  a )  X.  (
a  X.  a ) ) )  We  (
a  X.  a ) )
2927, 28sylib 196 . . . . . . . . . . . . 13  |-  ( ph  ->  ( R  i^i  (
( a  X.  a
)  X.  ( a  X.  a ) ) )  We  ( a  X.  a ) )
30 infxpen.1 . . . . . . . . . . . . . 14  |-  Q  =  ( R  i^i  (
( a  X.  a
)  X.  ( a  X.  a ) ) )
31 weeq1 4703 . . . . . . . . . . . . . 14  |-  ( Q  =  ( R  i^i  ( ( a  X.  a )  X.  (
a  X.  a ) ) )  ->  ( Q  We  ( a  X.  a )  <->  ( R  i^i  ( ( a  X.  a )  X.  (
a  X.  a ) ) )  We  (
a  X.  a ) ) )
3230, 31ax-mp 5 . . . . . . . . . . . . 13  |-  ( Q  We  ( a  X.  a )  <->  ( R  i^i  ( ( a  X.  a )  X.  (
a  X.  a ) ) )  We  (
a  X.  a ) )
3329, 32sylibr 212 . . . . . . . . . . . 12  |-  ( ph  ->  Q  We  ( a  X.  a ) )
34 infxpen.4 . . . . . . . . . . . . 13  |-  J  = OrdIso
( Q ,  ( a  X.  a ) )
3534oiiso 7743 . . . . . . . . . . . 12  |-  ( ( ( a  X.  a
)  e.  _V  /\  Q  We  ( a  X.  a ) )  ->  J  Isom  _E  ,  Q  ( dom  J ,  ( a  X.  a ) ) )
3615, 33, 35sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  J  Isom  _E  ,  Q  ( dom  J ,  ( a  X.  a ) ) )
37 isof1o 6011 . . . . . . . . . . 11  |-  ( J 
Isom  _E  ,  Q  ( dom  J ,  ( a  X.  a ) )  ->  J : dom  J -1-1-onto-> ( a  X.  a
) )
38 f1ocnv 5648 . . . . . . . . . . 11  |-  ( J : dom  J -1-1-onto-> ( a  X.  a )  ->  `' J : ( a  X.  a ) -1-1-onto-> dom  J
)
39 f1of1 5635 . . . . . . . . . . 11  |-  ( `' J : ( a  X.  a ) -1-1-onto-> dom  J  ->  `' J : ( a  X.  a ) -1-1-> dom  J )
4036, 37, 38, 394syl 21 . . . . . . . . . 10  |-  ( ph  ->  `' J : ( a  X.  a ) -1-1-> dom  J )
41 f1f1orn 5647 . . . . . . . . . 10  |-  ( `' J : ( a  X.  a ) -1-1-> dom  J  ->  `' J :
( a  X.  a
)
-1-1-onto-> ran  `' J )
4215f1oen 7322 . . . . . . . . . 10  |-  ( `' J : ( a  X.  a ) -1-1-onto-> ran  `' J  ->  ( a  X.  a )  ~~  ran  `' J )
4340, 41, 423syl 20 . . . . . . . . 9  |-  ( ph  ->  ( a  X.  a
)  ~~  ran  `' J
)
44 f1ofn 5637 . . . . . . . . . . 11  |-  ( `' J : ( a  X.  a ) -1-1-onto-> dom  J  ->  `' J  Fn  (
a  X.  a ) )
4536, 37, 38, 444syl 21 . . . . . . . . . 10  |-  ( ph  ->  `' J  Fn  (
a  X.  a ) )
4636adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  J  Isom  _E  ,  Q  ( dom  J ,  ( a  X.  a ) ) )
4737, 38, 393syl 20 . . . . . . . . . . . . . . . . . 18  |-  ( J 
Isom  _E  ,  Q  ( dom  J ,  ( a  X.  a ) )  ->  `' J : ( a  X.  a ) -1-1-> dom  J
)
48 cnvimass 5184 . . . . . . . . . . . . . . . . . . 19  |-  ( `' Q " { w } )  C_  dom  Q
49 inss2 3566 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  i^i  ( ( a  X.  a )  X.  ( a  X.  a
) ) )  C_  ( ( a  X.  a )  X.  (
a  X.  a ) )
5030, 49eqsstri 3381 . . . . . . . . . . . . . . . . . . . . 21  |-  Q  C_  ( ( a  X.  a )  X.  (
a  X.  a ) )
51 dmss 5034 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Q 
C_  ( ( a  X.  a )  X.  ( a  X.  a
) )  ->  dom  Q 
C_  dom  ( (
a  X.  a )  X.  ( a  X.  a ) ) )
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  dom  Q  C_ 
dom  ( ( a  X.  a )  X.  ( a  X.  a
) )
53 dmxpid 5054 . . . . . . . . . . . . . . . . . . . 20  |-  dom  (
( a  X.  a
)  X.  ( a  X.  a ) )  =  ( a  X.  a )
5452, 53sseqtri 3383 . . . . . . . . . . . . . . . . . . 19  |-  dom  Q  C_  ( a  X.  a
)
5548, 54sstri 3360 . . . . . . . . . . . . . . . . . 18  |-  ( `' Q " { w } )  C_  (
a  X.  a )
56 f1ores 5650 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' J : ( a  X.  a ) -1-1-> dom  J  /\  ( `' Q " { w } ) 
C_  ( a  X.  a ) )  -> 
( `' J  |`  ( `' Q " { w } ) ) : ( `' Q " { w } ) -1-1-onto-> ( `' J " ( `' Q " { w } ) ) )
5747, 55, 56sylancl 662 . . . . . . . . . . . . . . . . 17  |-  ( J 
Isom  _E  ,  Q  ( dom  J ,  ( a  X.  a ) )  ->  ( `' J  |`  ( `' Q " { w } ) ) : ( `' Q " { w } ) -1-1-onto-> ( `' J "
( `' Q " { w } ) ) )
5815, 15xpex 6503 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  X.  a )  X.  ( a  X.  a ) )  e. 
_V
5958inex2 4429 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  i^i  ( ( a  X.  a )  X.  ( a  X.  a
) ) )  e. 
_V
6030, 59eqeltri 2508 . . . . . . . . . . . . . . . . . . . 20  |-  Q  e. 
_V
6160cnvex 6520 . . . . . . . . . . . . . . . . . . 19  |-  `' Q  e.  _V
62 imaexg 6510 . . . . . . . . . . . . . . . . . . 19  |-  ( `' Q  e.  _V  ->  ( `' Q " { w } )  e.  _V )
6361, 62ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  ( `' Q " { w } )  e.  _V
6463f1oen 7322 . . . . . . . . . . . . . . . . 17  |-  ( ( `' J  |`  ( `' Q " { w } ) ) : ( `' Q " { w } ) -1-1-onto-> ( `' J " ( `' Q " { w } ) )  -> 
( `' Q " { w } ) 
~~  ( `' J " ( `' Q " { w } ) ) )
6546, 57, 643syl 20 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' Q " { w } )  ~~  ( `' J " ( `' Q " { w } ) ) )
66 sseqin2 3564 . . . . . . . . . . . . . . . . . . 19  |-  ( ( `' Q " { w } )  C_  (
a  X.  a )  <-> 
( ( a  X.  a )  i^i  ( `' Q " { w } ) )  =  ( `' Q " { w } ) )
6755, 66mpbi 208 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  X.  a )  i^i  ( `' Q " { w } ) )  =  ( `' Q " { w } )
6867imaeq2i 5162 . . . . . . . . . . . . . . . . 17  |-  ( `' J " ( ( a  X.  a )  i^i  ( `' Q " { w } ) ) )  =  ( `' J " ( `' Q " { w } ) )
69 isocnv 6016 . . . . . . . . . . . . . . . . . . . 20  |-  ( J 
Isom  _E  ,  Q  ( dom  J ,  ( a  X.  a ) )  ->  `' J  Isom  Q ,  _E  (
( a  X.  a
) ,  dom  J
) )
7046, 69syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  `' J  Isom  Q ,  _E  ( ( a  X.  a ) ,  dom  J ) )
71 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  w  e.  ( a  X.  a
) )
72 isoini 6024 . . . . . . . . . . . . . . . . . . 19  |-  ( ( `' J  Isom  Q ,  _E  ( ( a  X.  a ) ,  dom  J )  /\  w  e.  ( a  X.  a
) )  ->  ( `' J " ( ( a  X.  a )  i^i  ( `' Q " { w } ) ) )  =  ( dom  J  i^i  ( `'  _E  " { ( `' J `  w ) } ) ) )
7370, 71, 72syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J " ( ( a  X.  a )  i^i  ( `' Q " { w } ) ) )  =  ( dom  J  i^i  ( `'  _E  " { ( `' J `  w ) } ) ) )
74 fvex 5696 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' J `  w )  e.  _V
7574epini 5194 . . . . . . . . . . . . . . . . . . . 20  |-  ( `'  _E  " { ( `' J `  w ) } )  =  ( `' J `  w )
7675ineq2i 3544 . . . . . . . . . . . . . . . . . . 19  |-  ( dom 
J  i^i  ( `'  _E  " { ( `' J `  w ) } ) )  =  ( dom  J  i^i  ( `' J `  w ) )
7734oicl 7735 . . . . . . . . . . . . . . . . . . . . 21  |-  Ord  dom  J
78 f1of 5636 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( `' J : ( a  X.  a ) -1-1-onto-> dom  J  ->  `' J : ( a  X.  a ) --> dom 
J )
7936, 37, 38, 784syl 21 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  `' J : ( a  X.  a ) --> dom 
J )
8079ffvelrnda 5838 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J `  w )  e.  dom  J )
81 ordelss 4730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Ord  dom  J  /\  ( `' J `  w )  e.  dom  J )  ->  ( `' J `  w )  C_  dom  J )
8277, 80, 81sylancr 663 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J `  w ) 
C_  dom  J )
83 dfss1 3550 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( `' J `  w ) 
C_  dom  J  <->  ( dom  J  i^i  ( `' J `  w ) )  =  ( `' J `  w ) )
8482, 83sylib 196 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( dom  J  i^i  ( `' J `  w ) )  =  ( `' J `  w ) )
8576, 84syl5eq 2482 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( dom  J  i^i  ( `'  _E  " { ( `' J `  w ) } ) )  =  ( `' J `  w ) )
8673, 85eqtrd 2470 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J " ( ( a  X.  a )  i^i  ( `' Q " { w } ) ) )  =  ( `' J `  w ) )
8768, 86syl5eqr 2484 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J " ( `' Q " { w } ) )  =  ( `' J `  w ) )
8865, 87breqtrd 4311 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' Q " { w } )  ~~  ( `' J `  w ) )
8988ensymd 7352 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J `  w ) 
~~  ( `' Q " { w } ) )
90 infxpen.3 . . . . . . . . . . . . . . . . . . 19  |-  M  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )
91 fvex 5696 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1st `  w )  e.  _V
92 fvex 5696 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  w )  e.  _V
9391, 92unex 6373 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  w )  u.  ( 2nd `  w
) )  e.  _V
9490, 93eqeltri 2508 . . . . . . . . . . . . . . . . . 18  |-  M  e. 
_V
9594sucex 6417 . . . . . . . . . . . . . . . . 17  |-  suc  M  e.  _V
9695, 95xpex 6503 . . . . . . . . . . . . . . . 16  |-  ( suc 
M  X.  suc  M
)  e.  _V
97 xpss 4941 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  X.  a )  C_  ( _V  X.  _V )
98 simp3 990 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
z  e.  ( `' Q " { w } ) )
99 vex 2970 . . . . . . . . . . . . . . . . . . . . . . 23  |-  w  e. 
_V
100 vex 2970 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  z  e. 
_V
101100eliniseg 5193 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( w  e.  _V  ->  (
z  e.  ( `' Q " { w } )  <->  z Q w ) )
10299, 101ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z  e.  ( `' Q " { w } )  <-> 
z Q w )
10398, 102sylib 196 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
z Q w )
10430breqi 4293 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z Q w  <->  z ( R  i^i  ( ( a  X.  a )  X.  ( a  X.  a
) ) ) w )
105 brin 4336 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z ( R  i^i  (
( a  X.  a
)  X.  ( a  X.  a ) ) ) w  <->  ( z R w  /\  z
( ( a  X.  a )  X.  (
a  X.  a ) ) w ) )
106104, 105bitri 249 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z Q w  <->  ( z R w  /\  z
( ( a  X.  a )  X.  (
a  X.  a ) ) w ) )
107106simprbi 464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z Q w  ->  z
( ( a  X.  a )  X.  (
a  X.  a ) ) w )
108 brxp 4865 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z ( ( a  X.  a )  X.  (
a  X.  a ) ) w  <->  ( z  e.  ( a  X.  a
)  /\  w  e.  ( a  X.  a
) ) )
109108simplbi 460 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z ( ( a  X.  a )  X.  (
a  X.  a ) ) w  ->  z  e.  ( a  X.  a
) )
110103, 107, 1093syl 20 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
z  e.  ( a  X.  a ) )
11197, 110sseldi 3349 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
z  e.  ( _V 
X.  _V ) )
11217adantr 465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  a  e.  On )
1131123adant3 1008 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
a  e.  On )
114 xp1st 6601 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z  e.  ( a  X.  a )  ->  ( 1st `  z )  e.  a )
115 onelon 4739 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  On  /\  ( 1st `  z )  e.  a )  -> 
( 1st `  z
)  e.  On )
116114, 115sylan2 474 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( a  e.  On  /\  z  e.  ( a  X.  a ) )  -> 
( 1st `  z
)  e.  On )
117113, 110, 116syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
( 1st `  z
)  e.  On )
118 eloni 4724 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a  e.  On  ->  Ord  a )
119 elxp7 6604 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( w  e.  ( a  X.  a )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  a  /\  ( 2nd `  w
)  e.  a ) ) )
120119simprbi 464 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( w  e.  ( a  X.  a )  ->  (
( 1st `  w
)  e.  a  /\  ( 2nd `  w )  e.  a ) )
121 ordunel 6433 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( Ord  a  /\  ( 1st `  w )  e.  a  /\  ( 2nd `  w )  e.  a )  ->  ( ( 1st `  w )  u.  ( 2nd `  w
) )  e.  a )
1221213expib 1190 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( Ord  a  ->  ( (
( 1st `  w
)  e.  a  /\  ( 2nd `  w )  e.  a )  -> 
( ( 1st `  w
)  u.  ( 2nd `  w ) )  e.  a ) )
123118, 120, 122syl2im 38 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( a  e.  On  ->  (
w  e.  ( a  X.  a )  -> 
( ( 1st `  w
)  u.  ( 2nd `  w ) )  e.  a ) )
124112, 71, 123sylc 60 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  (
( 1st `  w
)  u.  ( 2nd `  w ) )  e.  a )
12590, 124syl5eqel 2522 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  M  e.  a )
126 simprr 756 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  ->  A. m  e.  a  m  ~<  a )
12713, 126sylbi 195 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  A. m  e.  a  m  ~<  a )
128 simprl 755 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  ->  om  C_  a
)
12913, 128sylbi 195 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  om  C_  a )
130 iscard 8137 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
card `  a )  =  a  <->  ( a  e.  On  /\  A. m  e.  a  m  ~<  a ) )
131 cardlim 8134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( om  C_  ( card `  a
)  <->  Lim  ( card `  a
) )
132 sseq2 3373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
card `  a )  =  a  ->  ( om  C_  ( card `  a
)  <->  om  C_  a )
)
133 limeq 4726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
card `  a )  =  a  ->  ( Lim  ( card `  a
)  <->  Lim  a ) )
134132, 133bibi12d 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
card `  a )  =  a  ->  ( ( om  C_  ( card `  a )  <->  Lim  ( card `  a ) )  <->  ( om  C_  a  <->  Lim  a ) ) )
135131, 134mpbii 211 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
card `  a )  =  a  ->  ( om  C_  a  <->  Lim  a ) )
136130, 135sylbir 213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( a  e.  On  /\  A. m  e.  a  m 
~<  a )  ->  ( om  C_  a  <->  Lim  a ) )
137136biimpa 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  m  ~<  a )  /\  om  C_  a )  ->  Lim  a )
13817, 127, 129, 137syl21anc 1217 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  Lim  a )
139138adantr 465 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  Lim  a )
140 limsuc 6455 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Lim  a  ->  ( M  e.  a  <->  suc  M  e.  a ) )
141139, 140syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( M  e.  a  <->  suc  M  e.  a ) )
142125, 141mpbid 210 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  suc  M  e.  a )
143 onelon 4739 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  On  /\  suc  M  e.  a )  ->  suc  M  e.  On )
144112, 142, 143syl2anc 661 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  suc  M  e.  On )
1451443adant3 1008 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  ->  suc  M  e.  On )
146 ssun1 3514 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1st `  z )  C_  (
( 1st `  z
)  u.  ( 2nd `  z ) )
147146a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
( 1st `  z
)  C_  ( ( 1st `  z )  u.  ( 2nd `  z
) ) )
148106simplbi 460 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z Q w  ->  z R w )
149 df-br 4288 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z R w  <->  <. z ,  w >.  e.  R
)
15023eleq2i 2502 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( <.
z ,  w >.  e.  R  <->  <. z ,  w >.  e.  { <. z ,  w >.  |  (
( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) } )
151 opabid 4591 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( <.
z ,  w >.  e. 
{ <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }  <-> 
( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) )
152149, 150, 1513bitri 271 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( z R w  <->  ( (
z  e.  ( On 
X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w )  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z
) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  /\  z L w ) ) ) )
153152simprbi 464 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z R w  ->  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) )
154 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  /\  z L w )  -> 
( ( 1st `  z
)  u.  ( 2nd `  z ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
155154orim2i 518 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) )  ->  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( 1st `  z
)  u.  ( 2nd `  z ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) ) )
156153, 155syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z R w  ->  (
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w
)  u.  ( 2nd `  w ) )  \/  ( ( 1st `  z
)  u.  ( 2nd `  z ) )  =  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) ) )
157 fvex 5696 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 1st `  z )  e.  _V
158 fvex 5696 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 2nd `  z )  e.  _V
159157, 158unex 6373 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( 1st `  z )  u.  ( 2nd `  z
) )  e.  _V
160159elsuc 4783 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e. 
suc  ( ( 1st `  w )  u.  ( 2nd `  w ) )  <-> 
( ( ( 1st `  z )  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w )  u.  ( 2nd `  w ) )  \/  ( ( 1st `  z )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) ) ) )
161156, 160sylibr 212 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z R w  ->  (
( 1st `  z
)  u.  ( 2nd `  z ) )  e. 
suc  ( ( 1st `  w )  u.  ( 2nd `  w ) ) )
162 suceq 4779 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( M  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  ->  suc  M  =  suc  ( ( 1st `  w
)  u.  ( 2nd `  w ) ) )
16390, 162ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22  |-  suc  M  =  suc  ( ( 1st `  w )  u.  ( 2nd `  w ) )
164161, 163syl6eleqr 2529 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z R w  ->  (
( 1st `  z
)  u.  ( 2nd `  z ) )  e. 
suc  M )
165103, 148, 1643syl 20 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
( ( 1st `  z
)  u.  ( 2nd `  z ) )  e. 
suc  M )
166 ontr2 4761 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 1st `  z
)  e.  On  /\  suc  M  e.  On )  ->  ( ( ( 1st `  z ) 
C_  ( ( 1st `  z )  u.  ( 2nd `  z ) )  /\  ( ( 1st `  z )  u.  ( 2nd `  z ) )  e.  suc  M )  ->  ( 1st `  z
)  e.  suc  M
) )
167166imp 429 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 1st `  z
)  e.  On  /\  suc  M  e.  On )  /\  ( ( 1st `  z )  C_  (
( 1st `  z
)  u.  ( 2nd `  z ) )  /\  ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e. 
suc  M ) )  ->  ( 1st `  z
)  e.  suc  M
)
168117, 145, 147, 165, 167syl22anc 1219 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
( 1st `  z
)  e.  suc  M
)
169 xp2nd 6602 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z  e.  ( a  X.  a )  ->  ( 2nd `  z )  e.  a )
170 onelon 4739 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  On  /\  ( 2nd `  z )  e.  a )  -> 
( 2nd `  z
)  e.  On )
171169, 170sylan2 474 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( a  e.  On  /\  z  e.  ( a  X.  a ) )  -> 
( 2nd `  z
)  e.  On )
172113, 110, 171syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
( 2nd `  z
)  e.  On )
173 ssun2 3515 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 2nd `  z )  C_  (
( 1st `  z
)  u.  ( 2nd `  z ) )
174173a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
( 2nd `  z
)  C_  ( ( 1st `  z )  u.  ( 2nd `  z
) ) )
175 ontr2 4761 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 2nd `  z
)  e.  On  /\  suc  M  e.  On )  ->  ( ( ( 2nd `  z ) 
C_  ( ( 1st `  z )  u.  ( 2nd `  z ) )  /\  ( ( 1st `  z )  u.  ( 2nd `  z ) )  e.  suc  M )  ->  ( 2nd `  z
)  e.  suc  M
) )
176175imp 429 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 2nd `  z
)  e.  On  /\  suc  M  e.  On )  /\  ( ( 2nd `  z )  C_  (
( 1st `  z
)  u.  ( 2nd `  z ) )  /\  ( ( 1st `  z
)  u.  ( 2nd `  z ) )  e. 
suc  M ) )  ->  ( 2nd `  z
)  e.  suc  M
)
177172, 145, 174, 165, 176syl22anc 1219 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
( 2nd `  z
)  e.  suc  M
)
178 elxp7 6604 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  ( suc  M  X.  suc  M )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  suc  M  /\  ( 2nd `  z
)  e.  suc  M
) ) )
179178biimpri 206 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  ( _V 
X.  _V )  /\  (
( 1st `  z
)  e.  suc  M  /\  ( 2nd `  z
)  e.  suc  M
) )  ->  z  e.  ( suc  M  X.  suc  M ) )
180111, 168, 177, 179syl12anc 1216 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  ( a  X.  a
)  /\  z  e.  ( `' Q " { w } ) )  -> 
z  e.  ( suc 
M  X.  suc  M
) )
1811803expia 1189 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  (
z  e.  ( `' Q " { w } )  ->  z  e.  ( suc  M  X.  suc  M ) ) )
182181ssrdv 3357 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' Q " { w } )  C_  ( suc  M  X.  suc  M
) )
183 ssdomg 7347 . . . . . . . . . . . . . . . 16  |-  ( ( suc  M  X.  suc  M )  e.  _V  ->  ( ( `' Q " { w } ) 
C_  ( suc  M  X.  suc  M )  -> 
( `' Q " { w } )  ~<_  ( suc  M  X.  suc  M ) ) )
18496, 182, 183mpsyl 63 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' Q " { w } )  ~<_  ( suc 
M  X.  suc  M
) )
185129adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  om  C_  a
)
186 nnfi 7495 . . . . . . . . . . . . . . . . . . . 20  |-  ( suc 
M  e.  om  ->  suc 
M  e.  Fin )
187 xpfi 7575 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( suc  M  e.  Fin  /\ 
suc  M  e.  Fin )  ->  ( suc  M  X.  suc  M )  e. 
Fin )
188187anidms 645 . . . . . . . . . . . . . . . . . . . . 21  |-  ( suc 
M  e.  Fin  ->  ( suc  M  X.  suc  M )  e.  Fin )
189 isfinite 7850 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( suc  M  X.  suc  M )  e.  Fin  <->  ( suc  M  X.  suc  M ) 
~<  om )
190188, 189sylib 196 . . . . . . . . . . . . . . . . . . . 20  |-  ( suc 
M  e.  Fin  ->  ( suc  M  X.  suc  M )  ~<  om )
191186, 190syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( suc 
M  e.  om  ->  ( suc  M  X.  suc  M )  ~<  om )
192 ssdomg 7347 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  e.  _V  ->  ( om  C_  a  ->  om  ~<_  a ) )
19314, 192ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  ( om  C_  a  ->  om  ~<_  a )
194 sdomdomtr 7436 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( suc  M  X.  suc  M )  ~<  om  /\  om  ~<_  a )  ->  ( suc  M  X.  suc  M
)  ~<  a )
195191, 193, 194syl2an 477 . . . . . . . . . . . . . . . . . 18  |-  ( ( suc  M  e.  om  /\ 
om  C_  a )  -> 
( suc  M  X.  suc  M )  ~<  a
)
196195expcom 435 . . . . . . . . . . . . . . . . 17  |-  ( om  C_  a  ->  ( suc 
M  e.  om  ->  ( suc  M  X.  suc  M )  ~<  a )
)
197185, 196syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( suc  M  e.  om  ->  ( suc  M  X.  suc  M )  ~<  a )
)
198127adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  A. m  e.  a  m  ~<  a )
199 breq1 4290 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  suc  M  -> 
( m  ~<  a  <->  suc 
M  ~<  a ) )
200199rspccv 3065 . . . . . . . . . . . . . . . . . 18  |-  ( A. m  e.  a  m  ~<  a  ->  ( suc  M  e.  a  ->  suc  M 
~<  a ) )
201198, 142, 200sylc 60 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  suc  M 
~<  a )
202 omelon 7844 . . . . . . . . . . . . . . . . . . 19  |-  om  e.  On
203 ontri1 4748 . . . . . . . . . . . . . . . . . . 19  |-  ( ( om  e.  On  /\  suc  M  e.  On )  ->  ( om  C_  suc  M  <->  -.  suc  M  e.  om ) )
204202, 144, 203sylancr 663 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( om  C_  suc  M  <->  -.  suc  M  e.  om ) )
205 simplr 754 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  ->  A. m  e.  a  ( om  C_  m  ->  ( m  X.  m )  ~~  m
) )
20613, 205sylbi 195 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)
207206adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  A. m  e.  a  ( om  C_  m  ->  ( m  X.  m )  ~~  m
) )
208 sseq2 3373 . . . . . . . . . . . . . . . . . . . . 21  |-  ( m  =  suc  M  -> 
( om  C_  m  <->  om  C_  suc  M ) )
209 xpeq12 4854 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( m  =  suc  M  /\  m  =  suc  M )  ->  ( m  X.  m )  =  ( suc  M  X.  suc  M ) )
210209anidms 645 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( m  =  suc  M  -> 
( m  X.  m
)  =  ( suc 
M  X.  suc  M
) )
211 id 22 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( m  =  suc  M  ->  m  =  suc  M )
212210, 211breq12d 4300 . . . . . . . . . . . . . . . . . . . . 21  |-  ( m  =  suc  M  -> 
( ( m  X.  m )  ~~  m  <->  ( suc  M  X.  suc  M )  ~~  suc  M
) )
213208, 212imbi12d 320 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  suc  M  -> 
( ( om  C_  m  ->  ( m  X.  m
)  ~~  m )  <->  ( om  C_  suc  M  -> 
( suc  M  X.  suc  M )  ~~  suc  M ) ) )
214213rspccv 3065 . . . . . . . . . . . . . . . . . . 19  |-  ( A. m  e.  a  ( om  C_  m  ->  (
m  X.  m ) 
~~  m )  -> 
( suc  M  e.  a  ->  ( om  C_  suc  M  ->  ( suc  M  X.  suc  M )  ~~  suc  M ) ) )
215207, 142, 214sylc 60 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( om  C_  suc  M  -> 
( suc  M  X.  suc  M )  ~~  suc  M ) )
216204, 215sylbird 235 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( -.  suc  M  e.  om  ->  ( suc  M  X.  suc  M )  ~~  suc  M ) )
217 ensdomtr 7439 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( suc  M  X.  suc  M )  ~~  suc  M  /\  suc  M  ~<  a )  ->  ( suc  M  X.  suc  M ) 
~<  a )
218217expcom 435 . . . . . . . . . . . . . . . . 17  |-  ( suc 
M  ~<  a  ->  (
( suc  M  X.  suc  M )  ~~  suc  M  ->  ( suc  M  X.  suc  M )  ~< 
a ) )
219201, 216, 218sylsyld 56 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( -.  suc  M  e.  om  ->  ( suc  M  X.  suc  M )  ~<  a
) )
220197, 219pm2.61d 158 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( suc  M  X.  suc  M
)  ~<  a )
221 domsdomtr 7438 . . . . . . . . . . . . . . 15  |-  ( ( ( `' Q " { w } )  ~<_  ( suc  M  X.  suc  M )  /\  ( suc  M  X.  suc  M
)  ~<  a )  -> 
( `' Q " { w } ) 
~<  a )
222184, 220, 221syl2anc 661 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' Q " { w } )  ~<  a
)
223 ensdomtr 7439 . . . . . . . . . . . . . 14  |-  ( ( ( `' J `  w )  ~~  ( `' Q " { w } )  /\  ( `' Q " { w } )  ~<  a
)  ->  ( `' J `  w )  ~<  a )
22489, 222, 223syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J `  w ) 
~<  a )
225 ordelon 4738 . . . . . . . . . . . . . . 15  |-  ( ( Ord  dom  J  /\  ( `' J `  w )  e.  dom  J )  ->  ( `' J `  w )  e.  On )
22677, 80, 225sylancr 663 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J `  w )  e.  On )
227 onenon 8111 . . . . . . . . . . . . . . 15  |-  ( a  e.  On  ->  a  e.  dom  card )
228112, 227syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  a  e.  dom  card )
229 cardsdomel 8136 . . . . . . . . . . . . . 14  |-  ( ( ( `' J `  w )  e.  On  /\  a  e.  dom  card )  ->  ( ( `' J `  w ) 
~<  a  <->  ( `' J `  w )  e.  (
card `  a )
) )
230226, 228, 229syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  (
( `' J `  w )  ~<  a  <->  ( `' J `  w )  e.  ( card `  a
) ) )
231224, 230mpbid 210 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J `  w )  e.  ( card `  a
) )
232 eleq2 2499 . . . . . . . . . . . . . 14  |-  ( (
card `  a )  =  a  ->  ( ( `' J `  w )  e.  ( card `  a
)  <->  ( `' J `  w )  e.  a ) )
233130, 232sylbir 213 . . . . . . . . . . . . 13  |-  ( ( a  e.  On  /\  A. m  e.  a  m 
~<  a )  ->  (
( `' J `  w )  e.  (
card `  a )  <->  ( `' J `  w )  e.  a ) )
234112, 198, 233syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  (
( `' J `  w )  e.  (
card `  a )  <->  ( `' J `  w )  e.  a ) )
235231, 234mpbid 210 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( a  X.  a
) )  ->  ( `' J `  w )  e.  a )
236235ralrimiva 2794 . . . . . . . . . 10  |-  ( ph  ->  A. w  e.  ( a  X.  a ) ( `' J `  w )  e.  a )
237 fnfvrnss 5866 . . . . . . . . . . 11  |-  ( ( `' J  Fn  (
a  X.  a )  /\  A. w  e.  ( a  X.  a
) ( `' J `  w )  e.  a )  ->  ran  `' J  C_  a )
238 ssdomg 7347 . . . . . . . . . . 11  |-  ( a  e.  _V  ->  ( ran  `' J  C_  a  ->  ran  `' J  ~<_  a )
)
23914, 237, 238mpsyl 63 . . . . . . . . . 10  |-  ( ( `' J  Fn  (
a  X.  a )  /\  A. w  e.  ( a  X.  a
) ( `' J `  w )  e.  a )  ->  ran  `' J  ~<_  a )
24045, 236, 239syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ran  `' J  ~<_  a )
241 endomtr 7359 . . . . . . . . 9  |-  ( ( ( a  X.  a
)  ~~  ran  `' J  /\  ran  `' J  ~<_  a )  ->  ( a  X.  a )  ~<_  a )
24243, 240, 241syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( a  X.  a
)  ~<_  a )
24313, 242sylbir 213 . . . . . . 7  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  ->  (
a  X.  a )  ~<_  a )
244 df1o2 6924 . . . . . . . . . . . 12  |-  1o  =  { (/) }
245 1onn 7070 . . . . . . . . . . . 12  |-  1o  e.  om
246244, 245eqeltrri 2509 . . . . . . . . . . 11  |-  { (/) }  e.  om
247 nnsdom 7851 . . . . . . . . . . 11  |-  ( {
(/) }  e.  om  ->  { (/) }  ~<  om )
248 sdomdom 7329 . . . . . . . . . . 11  |-  ( {
(/) }  ~<  om  ->  {
(/) }  ~<_  om )
249246, 247, 248mp2b 10 . . . . . . . . . 10  |-  { (/) }  ~<_  om
250 domtr 7354 . . . . . . . . . 10  |-  ( ( { (/) }  ~<_  om  /\  om  ~<_  a )  ->  { (/) }  ~<_  a )
251249, 193, 250sylancr 663 . . . . . . . . 9  |-  ( om  C_  a  ->  { (/) }  ~<_  a )
252 0ex 4417 . . . . . . . . . . . 12  |-  (/)  e.  _V
25314, 252xpsnen 7387 . . . . . . . . . . 11  |-  ( a  X.  { (/) } ) 
~~  a
254253ensymi 7351 . . . . . . . . . 10  |-  a  ~~  ( a  X.  { (/)
} )
25514xpdom2 7398 . . . . . . . . . 10  |-  ( {
(/) }  ~<_  a  ->  ( a  X.  { (/) } )  ~<_  ( a  X.  a ) )
256 endomtr 7359 . . . . . . . . . 10  |-  ( ( a  ~~  ( a  X.  { (/) } )  /\  ( a  X. 
{ (/) } )  ~<_  ( a  X.  a ) )  ->  a  ~<_  ( a  X.  a ) )
257254, 255, 256sylancr 663 . . . . . . . . 9  |-  ( {
(/) }  ~<_  a  ->  a  ~<_  ( a  X.  a
) )
258251, 257syl 16 . . . . . . . 8  |-  ( om  C_  a  ->  a  ~<_  ( a  X.  a ) )
259258ad2antrl 727 . . . . . . 7  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  ->  a  ~<_  ( a  X.  a
) )
260 sbth 7423 . . . . . . 7  |-  ( ( ( a  X.  a
)  ~<_  a  /\  a  ~<_  ( a  X.  a
) )  ->  (
a  X.  a ) 
~~  a )
261243, 259, 260syl2anc 661 . . . . . 6  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  A. m  e.  a  m  ~<  a ) )  ->  (
a  X.  a ) 
~~  a )
262261expr 615 . . . . 5  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  om  C_  a
)  ->  ( A. m  e.  a  m  ~<  a  ->  ( a  X.  a )  ~~  a
) )
263 simplr 754 . . . . . . . 8  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  ->  A. m  e.  a 
( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)
264 simpll 753 . . . . . . . . 9  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  -> 
a  e.  On )
265 simprr 756 . . . . . . . . 9  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  ->  -.  A. m  e.  a  m  ~<  a )
266 rexnal 2721 . . . . . . . . . 10  |-  ( E. m  e.  a  -.  m  ~<  a  <->  -.  A. m  e.  a  m  ~<  a )
267 onelss 4756 . . . . . . . . . . . . 13  |-  ( a  e.  On  ->  (
m  e.  a  ->  m  C_  a ) )
268 ssdomg 7347 . . . . . . . . . . . . 13  |-  ( a  e.  On  ->  (
m  C_  a  ->  m  ~<_  a ) )
269267, 268syld 44 . . . . . . . . . . . 12  |-  ( a  e.  On  ->  (
m  e.  a  ->  m  ~<_  a ) )
270 bren2 7332 . . . . . . . . . . . . 13  |-  ( m 
~~  a  <->  ( m  ~<_  a  /\  -.  m  ~<  a ) )
271270simplbi2 625 . . . . . . . . . . . 12  |-  ( m  ~<_  a  ->  ( -.  m  ~<  a  ->  m  ~~  a ) )
272269, 271syl6 33 . . . . . . . . . . 11  |-  ( a  e.  On  ->  (
m  e.  a  -> 
( -.  m  ~<  a  ->  m  ~~  a
) ) )
273272reximdvai 2821 . . . . . . . . . 10  |-  ( a  e.  On  ->  ( E. m  e.  a  -.  m  ~<  a  ->  E. m  e.  a  m  ~~  a ) )
274266, 273syl5bir 218 . . . . . . . . 9  |-  ( a  e.  On  ->  ( -.  A. m  e.  a  m  ~<  a  ->  E. m  e.  a  m 
~~  a ) )
275264, 265, 274sylc 60 . . . . . . . 8  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  ->  E. m  e.  a  m  ~~  a )
276 r19.29 2852 . . . . . . . 8  |-  ( ( A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )  /\  E. m  e.  a  m  ~~  a )  ->  E. m  e.  a  ( ( om  C_  m  ->  ( m  X.  m
)  ~~  m )  /\  m  ~~  a ) )
277263, 275, 276syl2anc 661 . . . . . . 7  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  ->  E. m  e.  a 
( ( om  C_  m  ->  ( m  X.  m
)  ~~  m )  /\  m  ~~  a ) )
278 simprl 755 . . . . . . . 8  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  ->  om  C_  a )
279 onelon 4739 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  On  /\  m  e.  a )  ->  m  e.  On )
280 ensym 7350 . . . . . . . . . . . . . . . . . 18  |-  ( m 
~~  a  ->  a  ~~  m )
281 domentr 7360 . . . . . . . . . . . . . . . . . 18  |-  ( ( om  ~<_  a  /\  a  ~~  m )  ->  om  ~<_  m )
282193, 280, 281syl2an 477 . . . . . . . . . . . . . . . . 17  |-  ( ( om  C_  a  /\  m  ~~  a )  ->  om 
~<_  m )
283 domnsym 7429 . . . . . . . . . . . . . . . . . . 19  |-  ( om  ~<_  m  ->  -.  m  ~<  om )
284 nnsdom 7851 . . . . . . . . . . . . . . . . . . 19  |-  ( m  e.  om  ->  m  ~<  om )
285283, 284nsyl 121 . . . . . . . . . . . . . . . . . 18  |-  ( om  ~<_  m  ->  -.  m  e.  om )
286 ontri1 4748 . . . . . . . . . . . . . . . . . . 19  |-  ( ( om  e.  On  /\  m  e.  On )  ->  ( om  C_  m  <->  -.  m  e.  om )
)
287202, 286mpan 670 . . . . . . . . . . . . . . . . . 18  |-  ( m  e.  On  ->  ( om  C_  m  <->  -.  m  e.  om ) )
288285, 287syl5ibr 221 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  On  ->  ( om 
~<_  m  ->  om  C_  m
) )
289279, 282, 288syl2im 38 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  On  /\  m  e.  a )  ->  ( ( om  C_  a  /\  m  ~~  a )  ->  om  C_  m ) )
290289expd 436 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  On  /\  m  e.  a )  ->  ( om  C_  a  ->  ( m  ~~  a  ->  om  C_  m )
) )
291290impcom 430 . . . . . . . . . . . . . 14  |-  ( ( om  C_  a  /\  ( a  e.  On  /\  m  e.  a ) )  ->  ( m  ~~  a  ->  om  C_  m
) )
292291imim1d 75 . . . . . . . . . . . . 13  |-  ( ( om  C_  a  /\  ( a  e.  On  /\  m  e.  a ) )  ->  ( ( om  C_  m  ->  (
m  X.  m ) 
~~  m )  -> 
( m  ~~  a  ->  ( m  X.  m
)  ~~  m )
) )
293292imp32 433 . . . . . . . . . . . 12  |-  ( ( ( om  C_  a  /\  ( a  e.  On  /\  m  e.  a ) )  /\  ( ( om  C_  m  ->  ( m  X.  m ) 
~~  m )  /\  m  ~~  a ) )  ->  ( m  X.  m )  ~~  m
)
294 entr 7353 . . . . . . . . . . . . . . . 16  |-  ( ( ( m  X.  m
)  ~~  m  /\  m  ~~  a )  -> 
( m  X.  m
)  ~~  a )
295294ancoms 453 . . . . . . . . . . . . . . 15  |-  ( ( m  ~~  a  /\  ( m  X.  m
)  ~~  m )  ->  ( m  X.  m
)  ~~  a )
296 xpen 7466 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  ~~  m  /\  a  ~~  m )  -> 
( a  X.  a
)  ~~  ( m  X.  m ) )
297296anidms 645 . . . . . . . . . . . . . . . . 17  |-  ( a 
~~  m  ->  (
a  X.  a ) 
~~  ( m  X.  m ) )
298 entr 7353 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  X.  a
)  ~~  ( m  X.  m )  /\  (
m  X.  m ) 
~~  a )  -> 
( a  X.  a
)  ~~  a )
299297, 298sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( a  ~~  m  /\  ( m  X.  m
)  ~~  a )  ->  ( a  X.  a
)  ~~  a )
300280, 299sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( m  ~~  a  /\  ( m  X.  m
)  ~~  a )  ->  ( a  X.  a
)  ~~  a )
301295, 300syldan 470 . . . . . . . . . . . . . 14  |-  ( ( m  ~~  a  /\  ( m  X.  m
)  ~~  m )  ->  ( a  X.  a
)  ~~  a )
302301ex 434 . . . . . . . . . . . . 13  |-  ( m 
~~  a  ->  (
( m  X.  m
)  ~~  m  ->  ( a  X.  a ) 
~~  a ) )
303302ad2antll 728 . . . . . . . . . . . 12  |-  ( ( ( om  C_  a  /\  ( a  e.  On  /\  m  e.  a ) )  /\  ( ( om  C_  m  ->  ( m  X.  m ) 
~~  m )  /\  m  ~~  a ) )  ->  ( ( m  X.  m )  ~~  m  ->  ( a  X.  a )  ~~  a
) )
304293, 303mpd 15 . . . . . . . . . . 11  |-  ( ( ( om  C_  a  /\  ( a  e.  On  /\  m  e.  a ) )  /\  ( ( om  C_  m  ->  ( m  X.  m ) 
~~  m )  /\  m  ~~  a ) )  ->  ( a  X.  a )  ~~  a
)
305304ex 434 . . . . . . . . . 10  |-  ( ( om  C_  a  /\  ( a  e.  On  /\  m  e.  a ) )  ->  ( (
( om  C_  m  ->  ( m  X.  m
)  ~~  m )  /\  m  ~~  a )  ->  ( a  X.  a )  ~~  a
) )
306305expr 615 . . . . . . . . 9  |-  ( ( om  C_  a  /\  a  e.  On )  ->  ( m  e.  a  ->  ( ( ( om  C_  m  ->  ( m  X.  m ) 
~~  m )  /\  m  ~~  a )  -> 
( a  X.  a
)  ~~  a )
) )
307306rexlimdv 2835 . . . . . . . 8  |-  ( ( om  C_  a  /\  a  e.  On )  ->  ( E. m  e.  a  ( ( om  C_  m  ->  ( m  X.  m )  ~~  m )  /\  m  ~~  a )  ->  (
a  X.  a ) 
~~  a ) )
308278, 264, 307syl2anc 661 . . . . . . 7  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  -> 
( E. m  e.  a  ( ( om  C_  m  ->  ( m  X.  m )  ~~  m )  /\  m  ~~  a )  ->  (
a  X.  a ) 
~~  a ) )
309277, 308mpd 15 . . . . . 6  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  ( om  C_  a  /\  -.  A. m  e.  a  m  ~<  a ) )  -> 
( a  X.  a
)  ~~  a )
310309expr 615 . . . . 5  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  om  C_  a
)  ->  ( -.  A. m  e.  a  m 
~<  a  ->  ( a  X.  a )  ~~  a ) )
311262, 310pm2.61d 158 . . . 4  |-  ( ( ( a  e.  On  /\ 
A. m  e.  a  ( om  C_  m  ->  ( m  X.  m
)  ~~  m )
)  /\  om  C_  a
)  ->  ( a  X.  a )  ~~  a
)
312311exp31 604 . . 3  |-  ( a  e.  On  ->  ( A. m  e.  a 
( om  C_  m  ->  ( m  X.  m
)  ~~  m )  ->  ( om  C_  a  ->  ( a  X.  a
)  ~~  a )
) )
3136, 12, 312tfis3 6463 . 2  |-  ( A  e.  On  ->  ( om  C_  A  ->  ( A  X.  A )  ~~  A ) )
314313imp 429 1  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   _Vcvv 2967    u. cun 3321    i^i cin 3322    C_ wss 3323   (/)c0 3632   {csn 3872   <.cop 3878   class class class wbr 4287   {copab 4344    _E cep 4625   Se wse 4672    We wwe 4673   Ord word 4713   Oncon0 4714   Lim wlim 4715   suc csuc 4716    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836    |` cres 4837   "cima 4838    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414   omcom 6471   1stc1st 6570   2ndc2nd 6571   1oc1o 6905    ~~ cen 7299    ~<_ cdom 7300    ~< csdm 7301   Fincfn 7302  OrdIsocoi 7715   cardccrd 8097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-oi 7716  df-card 8101
This theorem is referenced by:  infxpen  8173
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