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Theorem oaordex 6489
Description: Existence theorem for ordering of ordinal sum. Similar to Proposition 4.34(f) of [Mendelson] p. 266 and its converse. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
oaordex  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  E. x  e.  On  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem oaordex
StepHypRef Expression
1 onelss 4371 . . . . 5  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
21adantl 454 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  A  C_  B )
)
3 oawordex 6488 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  E. x  e.  On  ( A  +o  x )  =  B ) )
42, 3sylibd 207 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  E. x  e.  On  ( A  +o  x
)  =  B ) )
5 oaord1 6482 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
6 eleq2 2317 . . . . . . . . . . . . 13  |-  ( ( A  +o  x )  =  B  ->  ( A  e.  ( A  +o  x )  <->  A  e.  B ) )
75, 6sylan9bb 683 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  x  e.  On )  /\  ( A  +o  x )  =  B )  ->  ( (/)  e.  x  <->  A  e.  B ) )
87biimprcd 218 . . . . . . . . . . 11  |-  ( A  e.  B  ->  (
( ( A  e.  On  /\  x  e.  On )  /\  ( A  +o  x )  =  B )  ->  (/)  e.  x
) )
98exp4c 594 . . . . . . . . . 10  |-  ( A  e.  B  ->  ( A  e.  On  ->  ( x  e.  On  ->  ( ( A  +o  x
)  =  B  ->  (/) 
e.  x ) ) ) )
109com12 29 . . . . . . . . 9  |-  ( A  e.  On  ->  ( A  e.  B  ->  ( x  e.  On  ->  ( ( A  +o  x
)  =  B  ->  (/) 
e.  x ) ) ) )
1110imp4b 576 . . . . . . . 8  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( ( x  e.  On  /\  ( A  +o  x )  =  B )  ->  (/)  e.  x
) )
12 simpr 449 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( A  +o  x
)  =  B )  ->  ( A  +o  x )  =  B )
1312a1i 12 . . . . . . . 8  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( ( x  e.  On  /\  ( A  +o  x )  =  B )  ->  ( A  +o  x )  =  B ) )
1411, 13jcad 521 . . . . . . 7  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( ( x  e.  On  /\  ( A  +o  x )  =  B )  ->  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
1514exp3a 427 . . . . . 6  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( x  e.  On  ->  ( ( A  +o  x )  =  B  ->  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
1615reximdvai 2624 . . . . 5  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( E. x  e.  On  ( A  +o  x )  =  B  ->  E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
1716ex 425 . . . 4  |-  ( A  e.  On  ->  ( A  e.  B  ->  ( E. x  e.  On  ( A  +o  x
)  =  B  ->  E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
1817adantr 453 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  ( E. x  e.  On  ( A  +o  x )  =  B  ->  E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
194, 18mpdd 38 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
207biimpd 200 . . . . . . 7  |-  ( ( ( A  e.  On  /\  x  e.  On )  /\  ( A  +o  x )  =  B )  ->  ( (/)  e.  x  ->  A  e.  B ) )
2120exp31 590 . . . . . 6  |-  ( A  e.  On  ->  (
x  e.  On  ->  ( ( A  +o  x
)  =  B  -> 
( (/)  e.  x  ->  A  e.  B )
) ) )
2221com34 79 . . . . 5  |-  ( A  e.  On  ->  (
x  e.  On  ->  (
(/)  e.  x  ->  ( ( A  +o  x
)  =  B  ->  A  e.  B )
) ) )
2322imp4a 575 . . . 4  |-  ( A  e.  On  ->  (
x  e.  On  ->  ( ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) ) )
2423rexlimdv 2637 . . 3  |-  ( A  e.  On  ->  ( E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2524adantr 453 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2619, 25impbid 185 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  E. x  e.  On  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517    C_ wss 3094   (/)c0 3397   Oncon0 4329  (class class class)co 5757    +o coa 6409
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 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-recs 6321  df-rdg 6356  df-oadd 6416
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