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Theorem oawordex 6551
Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 6549 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 6549 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  E! x  e.  On  ( A  +o  x )  =  B )
21ex 423 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  E! x  e.  On  ( A  +o  x
)  =  B ) )
3 reurex 2755 . . 3  |-  ( E! x  e.  On  ( A  +o  x )  =  B  ->  E. x  e.  On  ( A  +o  x )  =  B )
42, 3syl6 29 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  E. x  e.  On  ( A  +o  x
)  =  B ) )
5 oawordexr 6550 . . . 4  |-  ( ( A  e.  On  /\  E. x  e.  On  ( A  +o  x )  =  B )  ->  A  C_  B )
65ex 423 . . 3  |-  ( A  e.  On  ->  ( E. x  e.  On  ( A  +o  x
)  =  B  ->  A  C_  B ) )
76adantr 451 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. x  e.  On  ( A  +o  x )  =  B  ->  A  C_  B
) )
84, 7impbid 183 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 176    /\ wa 358    = wceq 1623    e. wcel 1685   E.wrex 2545   E!wreu 2546    C_ wss 3153   Oncon0 4391  (class class class)co 5820    +o coa 6472
This theorem is referenced by:  oaordex  6552  oaass  6555  odi  6573  omeulem1  6576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-oadd 6479
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