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Theorem oawordeu 6999
Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)
Assertion
Ref Expression
oawordeu  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  E! x  e.  On  ( A  +o  x )  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem oawordeu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sseq1 3382 . . . 4  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  C_  B  <->  if ( A  e.  On ,  A ,  (/) )  C_  B ) )
2 oveq1 6103 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  +o  x )  =  ( if ( A  e.  On ,  A ,  (/) )  +o  x
) )
32eqeq1d 2451 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  (
( A  +o  x
)  =  B  <->  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B ) )
43reubidv 2910 . . . 4  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( E! x  e.  On  ( A  +o  x
)  =  B  <->  E! x  e.  On  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B ) )
51, 4imbi12d 320 . . 3  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  (
( A  C_  B  ->  E! x  e.  On  ( A  +o  x
)  =  B )  <-> 
( if ( A  e.  On ,  A ,  (/) )  C_  B  ->  E! x  e.  On  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B ) ) )
6 sseq2 3383 . . . 4  |-  ( B  =  if ( B  e.  On ,  B ,  (/) )  ->  ( if ( A  e.  On ,  A ,  (/) )  C_  B 
<->  if ( A  e.  On ,  A ,  (/) )  C_  if ( B  e.  On ,  B ,  (/) ) ) )
7 eqeq2 2452 . . . . 5  |-  ( B  =  if ( B  e.  On ,  B ,  (/) )  ->  (
( if ( A  e.  On ,  A ,  (/) )  +o  x
)  =  B  <->  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  if ( B  e.  On ,  B ,  (/) ) ) )
87reubidv 2910 . . . 4  |-  ( B  =  if ( B  e.  On ,  B ,  (/) )  ->  ( E! x  e.  On  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B  <->  E! x  e.  On  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  if ( B  e.  On ,  B ,  (/) ) ) )
96, 8imbi12d 320 . . 3  |-  ( B  =  if ( B  e.  On ,  B ,  (/) )  ->  (
( if ( A  e.  On ,  A ,  (/) )  C_  B  ->  E! x  e.  On  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B )  <->  ( if ( A  e.  On ,  A ,  (/) )  C_  if ( B  e.  On ,  B ,  (/) )  ->  E! x  e.  On  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  if ( B  e.  On ,  B ,  (/) ) ) ) )
10 0elon 4777 . . . . 5  |-  (/)  e.  On
1110elimel 3857 . . . 4  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1210elimel 3857 . . . 4  |-  if ( B  e.  On ,  B ,  (/) )  e.  On
13 eqid 2443 . . . 4  |-  { y  e.  On  |  if ( B  e.  On ,  B ,  (/) )  C_  ( if ( A  e.  On ,  A ,  (/) )  +o  y ) }  =  { y  e.  On  |  if ( B  e.  On ,  B ,  (/) )  C_  ( if ( A  e.  On ,  A ,  (/) )  +o  y ) }
1411, 12, 13oawordeulem 6998 . . 3  |-  ( if ( A  e.  On ,  A ,  (/) )  C_  if ( B  e.  On ,  B ,  (/) )  ->  E! x  e.  On  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  if ( B  e.  On ,  B ,  (/) ) )
155, 9, 14dedth2h 3847 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  E! x  e.  On  ( A  +o  x
)  =  B ) )
1615imp 429 1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  E! x  e.  On  ( A  +o  x )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E!wreu 2722   {crab 2724    C_ wss 3333   (/)c0 3642   ifcif 3796   Oncon0 4724  (class class class)co 6096    +o coa 6922
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-oadd 6929
This theorem is referenced by:  oawordex  7001  oaf1o  7007  oaabs  7088  oaabs2  7089
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