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Theorem oawordeulem 7252
Description: Lemma for oawordex 7255. (Contributed by NM, 11-Dec-2004.)
Hypotheses
Ref Expression
oawordeulem.1  |-  A  e.  On
oawordeulem.2  |-  B  e.  On
oawordeulem.3  |-  S  =  { y  e.  On  |  B  C_  ( A  +o  y ) }
Assertion
Ref Expression
oawordeulem  |-  ( A 
C_  B  ->  E! x  e.  On  ( A  +o  x )  =  B )
Distinct variable groups:    x, y, A    x, B, y    x, S
Allowed substitution hint:    S( y)

Proof of Theorem oawordeulem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 oawordeulem.3 . . . . . 6  |-  S  =  { y  e.  On  |  B  C_  ( A  +o  y ) }
2 ssrab2 3513 . . . . . 6  |-  { y  e.  On  |  B  C_  ( A  +o  y
) }  C_  On
31, 2eqsstri 3461 . . . . 5  |-  S  C_  On
4 oawordeulem.2 . . . . . . 7  |-  B  e.  On
5 oawordeulem.1 . . . . . . . 8  |-  A  e.  On
6 oaword2 7251 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
74, 5, 6mp2an 677 . . . . . . 7  |-  B  C_  ( A  +o  B
)
8 oveq2 6296 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  +o  y )  =  ( A  +o  B
) )
98sseq2d 3459 . . . . . . . 8  |-  ( y  =  B  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  B ) ) )
109, 1elrab2 3197 . . . . . . 7  |-  ( B  e.  S  <->  ( B  e.  On  /\  B  C_  ( A  +o  B
) ) )
114, 7, 10mpbir2an 930 . . . . . 6  |-  B  e.  S
1211ne0ii 3737 . . . . 5  |-  S  =/=  (/)
13 oninton 6624 . . . . 5  |-  ( ( S  C_  On  /\  S  =/=  (/) )  ->  |^| S  e.  On )
143, 12, 13mp2an 677 . . . 4  |-  |^| S  e.  On
15 onzsl 6670 . . . . . . . 8  |-  ( |^| S  e.  On  <->  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) ) )
1614, 15mpbi 212 . . . . . . 7  |-  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) )
17 oveq2 6296 . . . . . . . . . . 11  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  ( A  +o  (/) ) )
18 oa0 7215 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
195, 18ax-mp 5 . . . . . . . . . . 11  |-  ( A  +o  (/) )  =  A
2017, 19syl6eq 2500 . . . . . . . . . 10  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  A )
2120sseq1d 3458 . . . . . . . . 9  |-  ( |^| S  =  (/)  ->  (
( A  +o  |^| S )  C_  B  <->  A 
C_  B ) )
2221biimprd 227 . . . . . . . 8  |-  ( |^| S  =  (/)  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
23 oveq2 6296 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  -> 
( A  +o  |^| S )  =  ( A  +o  suc  z
) )
24 oasuc 7223 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z
) )
255, 24mpan 675 . . . . . . . . . . . 12  |-  ( z  e.  On  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z ) )
2623, 25sylan9eqr 2506 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  =  suc  ( A  +o  z ) )
27 vex 3047 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
2827sucid 5501 . . . . . . . . . . . . . 14  |-  z  e. 
suc  z
29 eleq2 2517 . . . . . . . . . . . . . 14  |-  ( |^| S  =  suc  z  -> 
( z  e.  |^| S 
<->  z  e.  suc  z
) )
3028, 29mpbiri 237 . . . . . . . . . . . . 13  |-  ( |^| S  =  suc  z  -> 
z  e.  |^| S
)
3114oneli 5529 . . . . . . . . . . . . . 14  |-  ( z  e.  |^| S  ->  z  e.  On )
321inteqi 4237 . . . . . . . . . . . . . . . . 17  |-  |^| S  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }
3332eleq2i 2520 . . . . . . . . . . . . . . . 16  |-  ( z  e.  |^| S  <->  z  e.  |^|
{ y  e.  On  |  B  C_  ( A  +o  y ) } )
34 oveq2 6296 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  ( A  +o  y )  =  ( A  +o  z
) )
3534sseq2d 3459 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  z ) ) )
3635onnminsb 6628 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  (
z  e.  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  -.  B  C_  ( A  +o  z ) ) )
3733, 36syl5bi 221 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  -.  B  C_  ( A  +o  z ) ) )
38 oacl 7234 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  z
)  e.  On )
395, 38mpan 675 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  On  ->  ( A  +o  z )  e.  On )
40 ontri1 5456 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  On  /\  ( A  +o  z
)  e.  On )  ->  ( B  C_  ( A  +o  z
)  <->  -.  ( A  +o  z )  e.  B
) )
414, 39, 40sylancr 668 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  ( B  C_  ( A  +o  z )  <->  -.  ( A  +o  z )  e.  B ) )
4241con2bid 331 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
( A  +o  z
)  e.  B  <->  -.  B  C_  ( A  +o  z
) ) )
4337, 42sylibrd 238 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  ( A  +o  z
)  e.  B ) )
4431, 43mpcom 37 . . . . . . . . . . . . 13  |-  ( z  e.  |^| S  ->  ( A  +o  z )  e.  B )
454onordi 5526 . . . . . . . . . . . . . 14  |-  Ord  B
46 ordsucss 6642 . . . . . . . . . . . . . 14  |-  ( Ord 
B  ->  ( ( A  +o  z )  e.  B  ->  suc  ( A  +o  z )  C_  B ) )
4745, 46ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( A  +o  z )  e.  B  ->  suc  ( A  +o  z
)  C_  B )
4830, 44, 473syl 18 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  ->  suc  ( A  +o  z
)  C_  B )
4948adantl 468 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  suc  ( A  +o  z )  C_  B )
5026, 49eqsstrd 3465 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  C_  B )
5150rexlimiva 2874 . . . . . . . . 9  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  +o  |^| S )  C_  B
)
5251a1d 26 . . . . . . . 8  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
53 oalim 7231 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  ( |^| S  e.  _V  /\ 
Lim  |^| S ) )  ->  ( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
545, 53mpan 675 . . . . . . . . . 10  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
55 iunss 4318 . . . . . . . . . . 11  |-  ( U_ z  e.  |^| S ( A  +o  z ) 
C_  B  <->  A. z  e.  |^| S ( A  +o  z )  C_  B )
564onelssi 5530 . . . . . . . . . . . 12  |-  ( ( A  +o  z )  e.  B  ->  ( A  +o  z )  C_  B )
5744, 56syl 17 . . . . . . . . . . 11  |-  ( z  e.  |^| S  ->  ( A  +o  z )  C_  B )
5855, 57mprgbir 2751 . . . . . . . . . 10  |-  U_ z  e.  |^| S ( A  +o  z )  C_  B
5954, 58syl6eqss 3481 . . . . . . . . 9  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  C_  B
)
6059a1d 26 . . . . . . . 8  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
6122, 52, 603jaoi 1330 . . . . . . 7  |-  ( (
|^| S  =  (/)  \/ 
E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e. 
_V  /\  Lim  |^| S
) )  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
6216, 61ax-mp 5 . . . . . 6  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  C_  B )
639rspcev 3149 . . . . . . . . 9  |-  ( ( B  e.  On  /\  B  C_  ( A  +o  B ) )  ->  E. y  e.  On  B  C_  ( A  +o  y ) )
644, 7, 63mp2an 677 . . . . . . . 8  |-  E. y  e.  On  B  C_  ( A  +o  y )
65 nfcv 2591 . . . . . . . . . 10  |-  F/_ y B
66 nfcv 2591 . . . . . . . . . . 11  |-  F/_ y A
67 nfcv 2591 . . . . . . . . . . 11  |-  F/_ y  +o
68 nfrab1 2970 . . . . . . . . . . . 12  |-  F/_ y { y  e.  On  |  B  C_  ( A  +o  y ) }
6968nfint 4243 . . . . . . . . . . 11  |-  F/_ y |^| { y  e.  On  |  B  C_  ( A  +o  y ) }
7066, 67, 69nfov 6314 . . . . . . . . . 10  |-  F/_ y
( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7165, 70nfss 3424 . . . . . . . . 9  |-  F/ y  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
72 oveq2 6296 . . . . . . . . . 10  |-  ( y  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  ( A  +o  y )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } ) )
7372sseq2d 3459 . . . . . . . . 9  |-  ( y  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } ) ) )
7471, 73onminsb 6623 . . . . . . . 8  |-  ( E. y  e.  On  B  C_  ( A  +o  y
)  ->  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } ) )
7564, 74ax-mp 5 . . . . . . 7  |-  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
7632oveq2i 6299 . . . . . . 7  |-  ( A  +o  |^| S )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7775, 76sseqtr4i 3464 . . . . . 6  |-  B  C_  ( A  +o  |^| S
)
7862, 77jctir 541 . . . . 5  |-  ( A 
C_  B  ->  (
( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
79 eqss 3446 . . . . 5  |-  ( ( A  +o  |^| S
)  =  B  <->  ( ( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
8078, 79sylibr 216 . . . 4  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  =  B )
81 oveq2 6296 . . . . . 6  |-  ( x  =  |^| S  -> 
( A  +o  x
)  =  ( A  +o  |^| S ) )
8281eqeq1d 2452 . . . . 5  |-  ( x  =  |^| S  -> 
( ( A  +o  x )  =  B  <-> 
( A  +o  |^| S )  =  B ) )
8382rspcev 3149 . . . 4  |-  ( (
|^| S  e.  On  /\  ( A  +o  |^| S )  =  B )  ->  E. x  e.  On  ( A  +o  x )  =  B )
8414, 80, 83sylancr 668 . . 3  |-  ( A 
C_  B  ->  E. x  e.  On  ( A  +o  x )  =  B )
85 eqtr3 2471 . . . . 5  |-  ( ( ( A  +o  x
)  =  B  /\  ( A  +o  y
)  =  B )  ->  ( A  +o  x )  =  ( A  +o  y ) )
86 oacan 7246 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  On  /\  y  e.  On )  ->  (
( A  +o  x
)  =  ( A  +o  y )  <->  x  =  y ) )
875, 86mp3an1 1350 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( A  +o  x )  =  ( A  +o  y )  <-> 
x  =  y ) )
8885, 87syl5ib 223 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) )
8988rgen2a 2814 . . 3  |-  A. x  e.  On  A. y  e.  On  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y )
9084, 89jctir 541 . 2  |-  ( A 
C_  B  ->  ( E. x  e.  On  ( A  +o  x
)  =  B  /\  A. x  e.  On  A. y  e.  On  (
( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) ) )
91 oveq2 6296 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
9291eqeq1d 2452 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  =  B  <->  ( A  +o  y )  =  B ) )
9392reu4 3231 . 2  |-  ( E! x  e.  On  ( A  +o  x )  =  B  <->  ( E. x  e.  On  ( A  +o  x )  =  B  /\  A. x  e.  On  A. y  e.  On  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) ) )
9490, 93sylibr 216 1  |-  ( A 
C_  B  ->  E! x  e.  On  ( A  +o  x )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    \/ w3o 983    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   E!wreu 2738   {crab 2740   _Vcvv 3044    C_ wss 3403   (/)c0 3730   |^|cint 4233   U_ciun 4277   Ord word 5421   Oncon0 5422   Lim wlim 5423   suc csuc 5424  (class class class)co 6288    +o coa 7176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-oadd 7183
This theorem is referenced by:  oawordeu  7253
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