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Theorem canthwe 8814
Description: The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 7460. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
Hypothesis
Ref Expression
canthwe.1  |-  O  =  { <. x ,  r
>.  |  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) }
Assertion
Ref Expression
canthwe  |-  ( A  e.  V  ->  A  ~<  O )
Distinct variable groups:    x, r, O    V, r, x    A, r, x

Proof of Theorem canthwe
Dummy variables  u  y  f  v  w  a  s  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 983 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
2 selpw 3864 . . . . . . . 8  |-  ( x  e.  ~P A  <->  x  C_  A
)
31, 2sylibr 212 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ~P A )
4 simp2 984 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( x  X.  x
) )
5 xpss12 4941 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  x  C_  A )  -> 
( x  X.  x
)  C_  ( A  X.  A ) )
61, 1, 5syl2anc 656 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  X.  x ) 
C_  ( A  X.  A ) )
74, 6sstrd 3363 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( A  X.  A
) )
8 selpw 3864 . . . . . . . 8  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
97, 8sylibr 212 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  e.  ~P ( A  X.  A ) )
103, 9jca 529 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) )
1110ssopab2i 4614 . . . . 5  |-  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }  C_  {
<. x ,  r >.  |  ( x  e. 
~P A  /\  r  e.  ~P ( A  X.  A ) ) }
12 canthwe.1 . . . . 5  |-  O  =  { <. x ,  r
>.  |  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) }
13 df-xp 4842 . . . . 5  |-  ( ~P A  X.  ~P ( A  X.  A ) )  =  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
1411, 12, 133sstr4i 3392 . . . 4  |-  O  C_  ( ~P A  X.  ~P ( A  X.  A
) )
15 pwexg 4473 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
16 xpexg 6506 . . . . . . 7  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1716anidms 640 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
18 pwexg 4473 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
1917, 18syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P ( A  X.  A
)  e.  _V )
20 xpexg 6506 . . . . 5  |-  ( ( ~P A  e.  _V  /\ 
~P ( A  X.  A )  e.  _V )  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )
2115, 19, 20syl2anc 656 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e.  _V )
22 ssexg 4435 . . . 4  |-  ( ( O  C_  ( ~P A  X.  ~P ( A  X.  A ) )  /\  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )  ->  O  e.  _V )
2314, 21, 22sylancr 658 . . 3  |-  ( A  e.  V  ->  O  e.  _V )
24 simpr 458 . . . . . . . 8  |-  ( ( A  e.  V  /\  u  e.  A )  ->  u  e.  A )
2524snssd 4015 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  ->  { u }  C_  A )
26 0ss 3663 . . . . . . . 8  |-  (/)  C_  ( { u }  X.  { u } )
2726a1i 11 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  C_  ( { u }  X.  { u }
) )
28 rel0 4960 . . . . . . . 8  |-  Rel  (/)
29 noel 3638 . . . . . . . . . 10  |-  -.  <. u ,  u >.  e.  (/)
30 df-br 4290 . . . . . . . . . 10  |-  ( u
(/) u  <->  <. u ,  u >.  e.  (/) )
3129, 30mtbir 299 . . . . . . . . 9  |-  -.  u (/) u
32 wesn 4906 . . . . . . . . 9  |-  ( Rel  (/)  ->  ( (/)  We  {
u }  <->  -.  u (/) u ) )
3331, 32mpbiri 233 . . . . . . . 8  |-  ( Rel  (/)  ->  (/)  We  { u } )
3428, 33mp1i 12 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  We  { u } )
35 snex 4530 . . . . . . . 8  |-  { u }  e.  _V
36 0ex 4419 . . . . . . . 8  |-  (/)  e.  _V
37 simpl 454 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  x  =  {
u } )
3837sseq1d 3380 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  C_  A 
<->  { u }  C_  A ) )
39 simpr 458 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  r  =  (/) )
4037, 37xpeq12d 4861 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  X.  x )  =  ( { u }  X.  { u } ) )
4139, 40sseq12d 3382 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  C_  ( x  X.  x
)  <->  (/)  C_  ( {
u }  X.  {
u } ) ) )
42 weeq2 4705 . . . . . . . . . 10  |-  ( x  =  { u }  ->  ( r  We  x  <->  r  We  { u }
) )
43 weeq1 4704 . . . . . . . . . 10  |-  ( r  =  (/)  ->  ( r  We  { u }  <->  (/)  We 
{ u } ) )
4442, 43sylan9bb 694 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  We  x  <->  (/)  We  { u } ) )
4538, 41, 443anbi123d 1284 . . . . . . . 8  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x )  <->  ( {
u }  C_  A  /\  (/)  C_  ( {
u }  X.  {
u } )  /\  (/) 
We  { u }
) ) )
4635, 36, 45opelopaba 4603 . . . . . . 7  |-  ( <. { u } ,  (/)
>.  e.  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }  <->  ( {
u }  C_  A  /\  (/)  C_  ( {
u }  X.  {
u } )  /\  (/) 
We  { u }
) )
4725, 27, 34, 46syl3anbrc 1167 . . . . . 6  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) } )
4847, 12syl6eleqr 2532 . . . . 5  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  O )
4948ex 434 . . . 4  |-  ( A  e.  V  ->  (
u  e.  A  ->  <. { u } ,  (/)
>.  e.  O ) )
50 eqid 2441 . . . . . . 7  |-  (/)  =  (/)
51 snex 4530 . . . . . . . 8  |-  { v }  e.  _V
5251, 36opth2 4567 . . . . . . 7  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  ( {
u }  =  {
v }  /\  (/)  =  (/) ) )
5350, 52mpbiran2 905 . . . . . 6  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  { u }  =  { v } )
54 vex 2973 . . . . . . 7  |-  u  e. 
_V
55 sneqbg 4040 . . . . . . 7  |-  ( u  e.  _V  ->  ( { u }  =  { v }  <->  u  =  v ) )
5654, 55ax-mp 5 . . . . . 6  |-  ( { u }  =  {
v }  <->  u  =  v )
5753, 56bitri 249 . . . . 5  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  u  =  v )
5857a1ii 27 . . . 4  |-  ( A  e.  V  ->  (
( u  e.  A  /\  v  e.  A
)  ->  ( <. { u } ,  (/) >.  =  <. { v } ,  (/) >.  <->  u  =  v
) ) )
5949, 58dom2d 7346 . . 3  |-  ( A  e.  V  ->  ( O  e.  _V  ->  A  ~<_  O ) )
6023, 59mpd 15 . 2  |-  ( A  e.  V  ->  A  ~<_  O )
61 eqid 2441 . . . . . . 7  |-  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }  =  { <. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) }
6261fpwwe2cbv 8793 . . . . . 6  |-  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }  =  { <. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  [. ( `' r " {
y } )  /  w ]. ( w f ( r  i^i  (
w  X.  w ) ) )  =  y ) ) }
63 eqid 2441 . . . . . 6  |-  U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) }  =  U. dom  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }
64 eqid 2441 . . . . . 6  |-  ( `' ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) " { ( U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) } f ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) ) } )  =  ( `' ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) " { ( U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) } f ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) ) } )
6512, 62, 63, 64canthwelem 8813 . . . . 5  |-  ( A  e.  V  ->  -.  f : O -1-1-> A )
66 f1of1 5637 . . . . 5  |-  ( f : O -1-1-onto-> A  ->  f : O -1-1-> A )
6765, 66nsyl 121 . . . 4  |-  ( A  e.  V  ->  -.  f : O -1-1-onto-> A )
6867nexdv 1939 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : O -1-1-onto-> A
)
69 ensym 7354 . . . 4  |-  ( A 
~~  O  ->  O  ~~  A )
70 bren 7315 . . . 4  |-  ( O 
~~  A  <->  E. f 
f : O -1-1-onto-> A )
7169, 70sylib 196 . . 3  |-  ( A 
~~  O  ->  E. f 
f : O -1-1-onto-> A )
7268, 71nsyl 121 . 2  |-  ( A  e.  V  ->  -.  A  ~~  O )
73 brsdom 7328 . 2  |-  ( A 
~<  O  <->  ( A  ~<_  O  /\  -.  A  ~~  O ) )
7460, 72, 73sylanbrc 659 1  |-  ( A  e.  V  ->  A  ~<  O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364   E.wex 1591    e. wcel 1761   A.wral 2713   _Vcvv 2970   [.wsbc 3183    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   {csn 3874   <.cop 3880   U.cuni 4088   class class class wbr 4289   {copab 4346    We wwe 4674    X. cxp 4834   `'ccnv 4835   dom cdm 4836   "cima 4839   Rel wrel 4841   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090    ~~ cen 7303    ~<_ cdom 7304    ~< csdm 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-recs 6828  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-oi 7720
This theorem is referenced by: (None)
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