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Theorem oawordeulem 6760
Description: Lemma for oawordex 6763. (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 3392 . . . . . 6  |-  { y  e.  On  |  B  C_  ( A  +o  y
) }  C_  On
31, 2eqsstri 3342 . . . . 5  |-  S  C_  On
4 oawordeulem.2 . . . . . . 7  |-  B  e.  On
5 oawordeulem.1 . . . . . . . 8  |-  A  e.  On
6 oaword2 6759 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
74, 5, 6mp2an 654 . . . . . . 7  |-  B  C_  ( A  +o  B
)
8 oveq2 6052 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  +o  y )  =  ( A  +o  B
) )
98sseq2d 3340 . . . . . . . 8  |-  ( y  =  B  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  B ) ) )
109, 1elrab2 3058 . . . . . . 7  |-  ( B  e.  S  <->  ( B  e.  On  /\  B  C_  ( A  +o  B
) ) )
114, 7, 10mpbir2an 887 . . . . . 6  |-  B  e.  S
12 ne0i 3598 . . . . . 6  |-  ( B  e.  S  ->  S  =/=  (/) )
1311, 12ax-mp 8 . . . . 5  |-  S  =/=  (/)
14 oninton 4743 . . . . 5  |-  ( ( S  C_  On  /\  S  =/=  (/) )  ->  |^| S  e.  On )
153, 13, 14mp2an 654 . . . 4  |-  |^| S  e.  On
16 onzsl 4789 . . . . . . . 8  |-  ( |^| S  e.  On  <->  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) ) )
1715, 16mpbi 200 . . . . . . 7  |-  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) )
18 oveq2 6052 . . . . . . . . . . 11  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  ( A  +o  (/) ) )
19 oa0 6723 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
205, 19ax-mp 8 . . . . . . . . . . 11  |-  ( A  +o  (/) )  =  A
2118, 20syl6eq 2456 . . . . . . . . . 10  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  A )
2221sseq1d 3339 . . . . . . . . 9  |-  ( |^| S  =  (/)  ->  (
( A  +o  |^| S )  C_  B  <->  A 
C_  B ) )
2322biimprd 215 . . . . . . . 8  |-  ( |^| S  =  (/)  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
24 oveq2 6052 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  -> 
( A  +o  |^| S )  =  ( A  +o  suc  z
) )
25 oasuc 6731 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z
) )
265, 25mpan 652 . . . . . . . . . . . 12  |-  ( z  e.  On  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z ) )
2724, 26sylan9eqr 2462 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  =  suc  ( A  +o  z ) )
28 vex 2923 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
2928sucid 4624 . . . . . . . . . . . . . 14  |-  z  e. 
suc  z
30 eleq2 2469 . . . . . . . . . . . . . 14  |-  ( |^| S  =  suc  z  -> 
( z  e.  |^| S 
<->  z  e.  suc  z
) )
3129, 30mpbiri 225 . . . . . . . . . . . . 13  |-  ( |^| S  =  suc  z  -> 
z  e.  |^| S
)
3215oneli 4652 . . . . . . . . . . . . . 14  |-  ( z  e.  |^| S  ->  z  e.  On )
331inteqi 4018 . . . . . . . . . . . . . . . . 17  |-  |^| S  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }
3433eleq2i 2472 . . . . . . . . . . . . . . . 16  |-  ( z  e.  |^| S  <->  z  e.  |^|
{ y  e.  On  |  B  C_  ( A  +o  y ) } )
35 oveq2 6052 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  ( A  +o  y )  =  ( A  +o  z
) )
3635sseq2d 3340 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  z ) ) )
3736onnminsb 4747 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  (
z  e.  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  -.  B  C_  ( A  +o  z ) ) )
3834, 37syl5bi 209 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  -.  B  C_  ( A  +o  z ) ) )
39 oacl 6742 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  z
)  e.  On )
405, 39mpan 652 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  On  ->  ( A  +o  z )  e.  On )
41 ontri1 4579 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  On  /\  ( A  +o  z
)  e.  On )  ->  ( B  C_  ( A  +o  z
)  <->  -.  ( A  +o  z )  e.  B
) )
424, 40, 41sylancr 645 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  ( B  C_  ( A  +o  z )  <->  -.  ( A  +o  z )  e.  B ) )
4342con2bid 320 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
( A  +o  z
)  e.  B  <->  -.  B  C_  ( A  +o  z
) ) )
4438, 43sylibrd 226 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  ( A  +o  z
)  e.  B ) )
4532, 44mpcom 34 . . . . . . . . . . . . 13  |-  ( z  e.  |^| S  ->  ( A  +o  z )  e.  B )
464onordi 4649 . . . . . . . . . . . . . 14  |-  Ord  B
47 ordsucss 4761 . . . . . . . . . . . . . 14  |-  ( Ord 
B  ->  ( ( A  +o  z )  e.  B  ->  suc  ( A  +o  z )  C_  B ) )
4846, 47ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( A  +o  z )  e.  B  ->  suc  ( A  +o  z
)  C_  B )
4931, 45, 483syl 19 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  ->  suc  ( A  +o  z
)  C_  B )
5049adantl 453 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  suc  ( A  +o  z )  C_  B )
5127, 50eqsstrd 3346 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  C_  B )
5251rexlimiva 2789 . . . . . . . . 9  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  +o  |^| S )  C_  B
)
5352a1d 23 . . . . . . . 8  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
54 oalim 6739 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  ( |^| S  e.  _V  /\ 
Lim  |^| S ) )  ->  ( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
555, 54mpan 652 . . . . . . . . . 10  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
56 iunss 4096 . . . . . . . . . . 11  |-  ( U_ z  e.  |^| S ( A  +o  z ) 
C_  B  <->  A. z  e.  |^| S ( A  +o  z )  C_  B )
574onelssi 4653 . . . . . . . . . . . 12  |-  ( ( A  +o  z )  e.  B  ->  ( A  +o  z )  C_  B )
5845, 57syl 16 . . . . . . . . . . 11  |-  ( z  e.  |^| S  ->  ( A  +o  z )  C_  B )
5956, 58mprgbir 2740 . . . . . . . . . 10  |-  U_ z  e.  |^| S ( A  +o  z )  C_  B
6055, 59syl6eqss 3362 . . . . . . . . 9  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  C_  B
)
6160a1d 23 . . . . . . . 8  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
6223, 53, 613jaoi 1247 . . . . . . 7  |-  ( (
|^| S  =  (/)  \/ 
E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e. 
_V  /\  Lim  |^| S
) )  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
6317, 62ax-mp 8 . . . . . 6  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  C_  B )
649rspcev 3016 . . . . . . . . 9  |-  ( ( B  e.  On  /\  B  C_  ( A  +o  B ) )  ->  E. y  e.  On  B  C_  ( A  +o  y ) )
654, 7, 64mp2an 654 . . . . . . . 8  |-  E. y  e.  On  B  C_  ( A  +o  y )
66 nfcv 2544 . . . . . . . . . 10  |-  F/_ y B
67 nfcv 2544 . . . . . . . . . . 11  |-  F/_ y A
68 nfcv 2544 . . . . . . . . . . 11  |-  F/_ y  +o
69 nfrab1 2852 . . . . . . . . . . . 12  |-  F/_ y { y  e.  On  |  B  C_  ( A  +o  y ) }
7069nfint 4024 . . . . . . . . . . 11  |-  F/_ y |^| { y  e.  On  |  B  C_  ( A  +o  y ) }
7167, 68, 70nfov 6067 . . . . . . . . . 10  |-  F/_ y
( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7266, 71nfss 3305 . . . . . . . . 9  |-  F/ y  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
73 oveq2 6052 . . . . . . . . . 10  |-  ( y  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  ( A  +o  y )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } ) )
7473sseq2d 3340 . . . . . . . . 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
) } ) ) )
7572, 74onminsb 4742 . . . . . . . 8  |-  ( E. y  e.  On  B  C_  ( A  +o  y
)  ->  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } ) )
7665, 75ax-mp 8 . . . . . . 7  |-  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
7733oveq2i 6055 . . . . . . 7  |-  ( A  +o  |^| S )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7876, 77sseqtr4i 3345 . . . . . 6  |-  B  C_  ( A  +o  |^| S
)
7963, 78jctir 525 . . . . 5  |-  ( A 
C_  B  ->  (
( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
80 eqss 3327 . . . . 5  |-  ( ( A  +o  |^| S
)  =  B  <->  ( ( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
8179, 80sylibr 204 . . . 4  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  =  B )
82 oveq2 6052 . . . . . 6  |-  ( x  =  |^| S  -> 
( A  +o  x
)  =  ( A  +o  |^| S ) )
8382eqeq1d 2416 . . . . 5  |-  ( x  =  |^| S  -> 
( ( A  +o  x )  =  B  <-> 
( A  +o  |^| S )  =  B ) )
8483rspcev 3016 . . . 4  |-  ( (
|^| S  e.  On  /\  ( A  +o  |^| S )  =  B )  ->  E. x  e.  On  ( A  +o  x )  =  B )
8515, 81, 84sylancr 645 . . 3  |-  ( A 
C_  B  ->  E. x  e.  On  ( A  +o  x )  =  B )
86 eqtr3 2427 . . . . 5  |-  ( ( ( A  +o  x
)  =  B  /\  ( A  +o  y
)  =  B )  ->  ( A  +o  x )  =  ( A  +o  y ) )
87 oacan 6754 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  On  /\  y  e.  On )  ->  (
( A  +o  x
)  =  ( A  +o  y )  <->  x  =  y ) )
885, 87mp3an1 1266 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( A  +o  x )  =  ( A  +o  y )  <-> 
x  =  y ) )
8986, 88syl5ib 211 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) )
9089rgen2a 2736 . . 3  |-  A. x  e.  On  A. y  e.  On  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y )
9185, 90jctir 525 . 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 ) ) )
92 oveq2 6052 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
9392eqeq1d 2416 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  =  B  <->  ( A  +o  y )  =  B ) )
9493reu4 3092 . 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 ) ) )
9591, 94sylibr 204 1  |-  ( A 
C_  B  ->  E! x  e.  On  ( A  +o  x )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   E.wrex 2671   E!wreu 2672   {crab 2674   _Vcvv 2920    C_ wss 3284   (/)c0 3592   |^|cint 4014   U_ciun 4057   Ord word 4544   Oncon0 4545   Lim wlim 4546   suc csuc 4547  (class class class)co 6044    +o coa 6684
This theorem is referenced by:  oawordeu  6761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-recs 6596  df-rdg 6631  df-oadd 6691
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