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Theorem oawordeulem 6552
Description: Lemma for oawordex 6555. (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 3258 . . . . . 6  |-  { y  e.  On  |  B  C_  ( A  +o  y
) }  C_  On
31, 2eqsstri 3208 . . . . 5  |-  S  C_  On
4 oawordeulem.2 . . . . . . 7  |-  B  e.  On
5 oawordeulem.1 . . . . . . . 8  |-  A  e.  On
6 oaword2 6551 . . . . . . . 8  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
74, 5, 6mp2an 653 . . . . . . 7  |-  B  C_  ( A  +o  B
)
8 oveq2 5866 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  +o  y )  =  ( A  +o  B
) )
98sseq2d 3206 . . . . . . . 8  |-  ( y  =  B  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  B ) ) )
109, 1elrab2 2925 . . . . . . 7  |-  ( B  e.  S  <->  ( B  e.  On  /\  B  C_  ( A  +o  B
) ) )
114, 7, 10mpbir2an 886 . . . . . 6  |-  B  e.  S
12 ne0i 3461 . . . . . 6  |-  ( B  e.  S  ->  S  =/=  (/) )
1311, 12ax-mp 8 . . . . 5  |-  S  =/=  (/)
14 oninton 4591 . . . . 5  |-  ( ( S  C_  On  /\  S  =/=  (/) )  ->  |^| S  e.  On )
153, 13, 14mp2an 653 . . . 4  |-  |^| S  e.  On
16 onzsl 4637 . . . . . . . 8  |-  ( |^| S  e.  On  <->  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) ) )
1715, 16mpbi 199 . . . . . . 7  |-  ( |^| S  =  (/)  \/  E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e.  _V  /\  Lim  |^| S ) )
18 oveq2 5866 . . . . . . . . . . 11  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  ( A  +o  (/) ) )
19 oa0 6515 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
205, 19ax-mp 8 . . . . . . . . . . 11  |-  ( A  +o  (/) )  =  A
2118, 20syl6eq 2331 . . . . . . . . . 10  |-  ( |^| S  =  (/)  ->  ( A  +o  |^| S )  =  A )
2221sseq1d 3205 . . . . . . . . 9  |-  ( |^| S  =  (/)  ->  (
( A  +o  |^| S )  C_  B  <->  A 
C_  B ) )
2322biimprd 214 . . . . . . . 8  |-  ( |^| S  =  (/)  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
24 oveq2 5866 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  -> 
( A  +o  |^| S )  =  ( A  +o  suc  z
) )
25 oasuc 6523 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z
) )
265, 25mpan 651 . . . . . . . . . . . 12  |-  ( z  e.  On  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z ) )
2724, 26sylan9eqr 2337 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  =  suc  ( A  +o  z ) )
28 vex 2791 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
2928sucid 4471 . . . . . . . . . . . . . 14  |-  z  e. 
suc  z
30 eleq2 2344 . . . . . . . . . . . . . 14  |-  ( |^| S  =  suc  z  -> 
( z  e.  |^| S 
<->  z  e.  suc  z
) )
3129, 30mpbiri 224 . . . . . . . . . . . . 13  |-  ( |^| S  =  suc  z  -> 
z  e.  |^| S
)
3215oneli 4500 . . . . . . . . . . . . . 14  |-  ( z  e.  |^| S  ->  z  e.  On )
331inteqi 3866 . . . . . . . . . . . . . . . . 17  |-  |^| S  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }
3433eleq2i 2347 . . . . . . . . . . . . . . . 16  |-  ( z  e.  |^| S  <->  z  e.  |^|
{ y  e.  On  |  B  C_  ( A  +o  y ) } )
35 oveq2 5866 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  ( A  +o  y )  =  ( A  +o  z
) )
3635sseq2d 3206 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  ( B  C_  ( A  +o  y )  <->  B  C_  ( A  +o  z ) ) )
3736onnminsb 4595 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  (
z  e.  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  -.  B  C_  ( A  +o  z ) ) )
3834, 37syl5bi 208 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  -.  B  C_  ( A  +o  z ) ) )
39 oacl 6534 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  On  /\  z  e.  On )  ->  ( A  +o  z
)  e.  On )
405, 39mpan 651 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  On  ->  ( A  +o  z )  e.  On )
41 ontri1 4426 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  On  /\  ( A  +o  z
)  e.  On )  ->  ( B  C_  ( A  +o  z
)  <->  -.  ( A  +o  z )  e.  B
) )
424, 40, 41sylancr 644 . . . . . . . . . . . . . . . 16  |-  ( z  e.  On  ->  ( B  C_  ( A  +o  z )  <->  -.  ( A  +o  z )  e.  B ) )
4342con2bid 319 . . . . . . . . . . . . . . 15  |-  ( z  e.  On  ->  (
( A  +o  z
)  e.  B  <->  -.  B  C_  ( A  +o  z
) ) )
4438, 43sylibrd 225 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
z  e.  |^| S  ->  ( A  +o  z
)  e.  B ) )
4532, 44mpcom 32 . . . . . . . . . . . . 13  |-  ( z  e.  |^| S  ->  ( A  +o  z )  e.  B )
464onordi 4497 . . . . . . . . . . . . . 14  |-  Ord  B
47 ordsucss 4609 . . . . . . . . . . . . . 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 18 . . . . . . . . . . . 12  |-  ( |^| S  =  suc  z  ->  suc  ( A  +o  z
)  C_  B )
5049adantl 452 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  suc  ( A  +o  z )  C_  B )
5127, 50eqsstrd 3212 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  |^| S  =  suc  z
)  ->  ( A  +o  |^| S )  C_  B )
5251rexlimiva 2662 . . . . . . . . 9  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  +o  |^| S )  C_  B
)
5352a1d 22 . . . . . . . 8  |-  ( E. z  e.  On  |^| S  =  suc  z  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
54 iunss 3943 . . . . . . . . . . 11  |-  ( U_ z  e.  |^| S ( A  +o  z ) 
C_  B  <->  A. z  e.  |^| S ( A  +o  z )  C_  B )
554onelssi 4501 . . . . . . . . . . . 12  |-  ( ( A  +o  z )  e.  B  ->  ( A  +o  z )  C_  B )
5645, 55syl 15 . . . . . . . . . . 11  |-  ( z  e.  |^| S  ->  ( A  +o  z )  C_  B )
5754, 56mprgbir 2613 . . . . . . . . . 10  |-  U_ z  e.  |^| S ( A  +o  z )  C_  B
58 oalim 6531 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  ( |^| S  e.  _V  /\ 
Lim  |^| S ) )  ->  ( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
595, 58mpan 651 . . . . . . . . . . 11  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  =  U_ z  e.  |^| S ( A  +o  z ) )
6059sseq1d 3205 . . . . . . . . . 10  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( ( A  +o  |^| S )  C_  B  <->  U_ z  e.  |^| S
( A  +o  z
)  C_  B )
)
6157, 60mpbiri 224 . . . . . . . . 9  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  +o  |^| S )  C_  B
)
6261a1d 22 . . . . . . . 8  |-  ( (
|^| S  e.  _V  /\ 
Lim  |^| S )  -> 
( A  C_  B  ->  ( A  +o  |^| S )  C_  B
) )
6323, 53, 623jaoi 1245 . . . . . . 7  |-  ( (
|^| S  =  (/)  \/ 
E. z  e.  On  |^| S  =  suc  z  \/  ( |^| S  e. 
_V  /\  Lim  |^| S
) )  ->  ( A  C_  B  ->  ( A  +o  |^| S )  C_  B ) )
6417, 63ax-mp 8 . . . . . 6  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  C_  B )
659rspcev 2884 . . . . . . . . 9  |-  ( ( B  e.  On  /\  B  C_  ( A  +o  B ) )  ->  E. y  e.  On  B  C_  ( A  +o  y ) )
664, 7, 65mp2an 653 . . . . . . . 8  |-  E. y  e.  On  B  C_  ( A  +o  y )
67 nfcv 2419 . . . . . . . . . 10  |-  F/_ y B
68 nfcv 2419 . . . . . . . . . . 11  |-  F/_ y A
69 nfcv 2419 . . . . . . . . . . 11  |-  F/_ y  +o
70 nfrab1 2720 . . . . . . . . . . . 12  |-  F/_ y { y  e.  On  |  B  C_  ( A  +o  y ) }
7170nfint 3872 . . . . . . . . . . 11  |-  F/_ y |^| { y  e.  On  |  B  C_  ( A  +o  y ) }
7268, 69, 71nfov 5881 . . . . . . . . . 10  |-  F/_ y
( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7367, 72nfss 3173 . . . . . . . . 9  |-  F/ y  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
74 oveq2 5866 . . . . . . . . . 10  |-  ( y  =  |^| { y  e.  On  |  B  C_  ( A  +o  y
) }  ->  ( A  +o  y )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } ) )
7574sseq2d 3206 . . . . . . . . 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
) } ) ) )
7673, 75onminsb 4590 . . . . . . . 8  |-  ( E. y  e.  On  B  C_  ( A  +o  y
)  ->  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } ) )
7766, 76ax-mp 8 . . . . . . 7  |-  B  C_  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y
) } )
7833oveq2i 5869 . . . . . . 7  |-  ( A  +o  |^| S )  =  ( A  +o  |^| { y  e.  On  |  B  C_  ( A  +o  y ) } )
7977, 78sseqtr4i 3211 . . . . . 6  |-  B  C_  ( A  +o  |^| S
)
8064, 79jctir 524 . . . . 5  |-  ( A 
C_  B  ->  (
( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
81 eqss 3194 . . . . 5  |-  ( ( A  +o  |^| S
)  =  B  <->  ( ( A  +o  |^| S )  C_  B  /\  B  C_  ( A  +o  |^| S ) ) )
8280, 81sylibr 203 . . . 4  |-  ( A 
C_  B  ->  ( A  +o  |^| S )  =  B )
83 oveq2 5866 . . . . . 6  |-  ( x  =  |^| S  -> 
( A  +o  x
)  =  ( A  +o  |^| S ) )
8483eqeq1d 2291 . . . . 5  |-  ( x  =  |^| S  -> 
( ( A  +o  x )  =  B  <-> 
( A  +o  |^| S )  =  B ) )
8584rspcev 2884 . . . 4  |-  ( (
|^| S  e.  On  /\  ( A  +o  |^| S )  =  B )  ->  E. x  e.  On  ( A  +o  x )  =  B )
8615, 82, 85sylancr 644 . . 3  |-  ( A 
C_  B  ->  E. x  e.  On  ( A  +o  x )  =  B )
87 eqtr3 2302 . . . . 5  |-  ( ( ( A  +o  x
)  =  B  /\  ( A  +o  y
)  =  B )  ->  ( A  +o  x )  =  ( A  +o  y ) )
88 oacan 6546 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  On  /\  y  e.  On )  ->  (
( A  +o  x
)  =  ( A  +o  y )  <->  x  =  y ) )
895, 88mp3an1 1264 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( A  +o  x )  =  ( A  +o  y )  <-> 
x  =  y ) )
9087, 89syl5ib 210 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y ) )
9190rgen2a 2609 . . 3  |-  A. x  e.  On  A. y  e.  On  ( ( ( A  +o  x )  =  B  /\  ( A  +o  y )  =  B )  ->  x  =  y )
9286, 91jctir 524 . 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 ) ) )
93 oveq2 5866 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
9493eqeq1d 2291 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  =  B  <->  ( A  +o  y )  =  B ) )
9594reu4 2959 . 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 ) ) )
9692, 95sylibr 203 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 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   |^|cint 3862   U_ciun 3905   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394  (class class class)co 5858    +o coa 6476
This theorem is referenced by:  oawordeu  6553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
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