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Theorem oawordex 6736
Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 6734 for uniqueness. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
oawordex  |-  ( ( 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 oawordex
StepHypRef Expression
1 oawordeu 6734 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  E! x  e.  On  ( A  +o  x )  =  B )
21ex 424 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  E! x  e.  On  ( A  +o  x
)  =  B ) )
3 reurex 2865 . . 3  |-  ( E! x  e.  On  ( A  +o  x )  =  B  ->  E. x  e.  On  ( A  +o  x )  =  B )
42, 3syl6 31 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  E. x  e.  On  ( A  +o  x
)  =  B ) )
5 oawordexr 6735 . . . 4  |-  ( ( A  e.  On  /\  E. x  e.  On  ( A  +o  x )  =  B )  ->  A  C_  B )
65ex 424 . . 3  |-  ( A  e.  On  ->  ( E. x  e.  On  ( A  +o  x
)  =  B  ->  A  C_  B ) )
76adantr 452 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. x  e.  On  ( A  +o  x )  =  B  ->  A  C_  B
) )
84, 7impbid 184 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    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2650   E!wreu 2651    C_ wss 3263   Oncon0 4522  (class class class)co 6020    +o coa 6657
This theorem is referenced by:  oaordex  6737  oaass  6740  odi  6758  omeulem1  6761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-recs 6569  df-rdg 6604  df-oadd 6664
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