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Theorem oawordeu 6507
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
StepHypRef Expression
1 sseq1 3160 . . . 4  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  C_  B  <->  if ( A  e.  On ,  A ,  (/) )  C_  B ) )
2 oveq1 5785 . . . . . 6  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  ( A  +o  x )  =  ( if ( A  e.  On ,  A ,  (/) )  +o  x
) )
32eqeq1d 2264 . . . . 5  |-  ( A  =  if ( A  e.  On ,  A ,  (/) )  ->  (
( A  +o  x
)  =  B  <->  ( if ( A  e.  On ,  A ,  (/) )  +o  x )  =  B ) )
43reubidv 2697 . . . 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 3161 . . . 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 2265 . . . . 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 2697 . . . 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 4403 . . . . 5  |-  (/)  e.  On
1110elimel 3577 . . . 4  |-  if ( A  e.  On ,  A ,  (/) )  e.  On
1210elimel 3577 . . . 4  |-  if ( B  e.  On ,  B ,  (/) )  e.  On
13 eqid 2256 . . . 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 6506 . . 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 3567 . 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 1619    e. wcel 1621   E!wreu 2518   {crab 2520    C_ wss 3113   (/)c0 3416   ifcif 3525   Oncon0 4350  (class class class)co 5778    +o coa 6430
This theorem is referenced by:  oawordex  6509  oaf1o  6515  oaabs  6596  oaabs2  6597
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-recs 6342  df-rdg 6377  df-oadd 6437
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