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Theorem oawordeu 7216
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 3530 . . . 4  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  C_  B  <->  if ( A  e.  On ,  A ,  (/) )  C_  B ) )
2 oveq1 6302 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  +o  x )  =  ( if ( A  e.  On ,  A ,  (/) )  +o  x
) )
32eqeq1d 2469 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  (
( A  +o  x
)  =  B  <->  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B ) )
43reubidv 3051 . . . 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 3531 . . . 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 2482 . . . . 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 3051 . . . 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 4937 . . . . 5  |-  (/)  e.  On
1110elimel 4008 . . . 4  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1210elimel 4008 . . . 4  |-  if ( B  e.  On ,  B ,  (/) )  e.  On
13 eqid 2467 . . . 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 7215 . . 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 3998 . 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 1379    e. wcel 1767   E!wreu 2819   {crab 2821    C_ wss 3481   (/)c0 3790   ifcif 3945   Oncon0 4884  (class class class)co 6295    +o coa 7139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-oadd 7146
This theorem is referenced by:  oawordex  7218  oaf1o  7224  oaabs  7305  oaabs2  7306
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