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Theorem oawordeu 6784
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 3356 . . . 4  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  C_  B  <->  if ( A  e.  On ,  A ,  (/) )  C_  B ) )
2 oveq1 6074 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  +o  x )  =  ( if ( A  e.  On ,  A ,  (/) )  +o  x
) )
32eqeq1d 2438 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  (
( A  +o  x
)  =  B  <->  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B ) )
43reubidv 2879 . . . 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 312 . . 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 3357 . . . 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 2439 . . . . 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 2879 . . . 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 312 . . 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 4621 . . . . 5  |-  (/)  e.  On
1110elimel 3778 . . . 4  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1210elimel 3778 . . . 4  |-  if ( B  e.  On ,  B ,  (/) )  e.  On
13 eqid 2430 . . . 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 6783 . . 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 3768 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  E! x  e.  On  ( A  +o  x
)  =  B ) )
1615imp 419 1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  E! x  e.  On  ( A  +o  x )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E!wreu 2694   {crab 2696    C_ wss 3307   (/)c0 3615   ifcif 3726   Oncon0 4568  (class class class)co 6067    +o coa 6707
This theorem is referenced by:  oawordex  6786  oaf1o  6792  oaabs  6873  oaabs2  6874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-recs 6619  df-rdg 6654  df-oadd 6714
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