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Theorem oawordeu 6549
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem oawordeu
StepHypRef Expression
1 sseq1 3201 . . . 4  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  C_  B  <->  if ( A  e.  On ,  A ,  (/) )  C_  B ) )
2 oveq1 5827 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  +o  x )  =  ( if ( A  e.  On ,  A ,  (/) )  +o  x
) )
32eqeq1d 2293 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  (
( A  +o  x
)  =  B  <->  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B ) )
43reubidv 2726 . . . 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 313 . . 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 3202 . . . 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 2294 . . . . 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 2726 . . . 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 313 . . 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 4445 . . . . 5  |-  (/)  e.  On
1110elimel 3619 . . . 4  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1210elimel 3619 . . . 4  |-  if ( B  e.  On ,  B ,  (/) )  e.  On
13 eqid 2285 . . . 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 6548 . . 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 3609 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  E! x  e.  On  ( A  +o  x
)  =  B ) )
1615imp 420 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 6    /\ wa 360    = wceq 1624    e. wcel 1685   E!wreu 2547   {crab 2549    C_ wss 3154   (/)c0 3457   ifcif 3567   Oncon0 4392  (class class class)co 5820    +o coa 6472
This theorem is referenced by:  oawordex  6551  oaf1o  6557  oaabs  6638  oaabs2  6639
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-oadd 6479
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