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Theorem oaordex 6442
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 4327 . . . . 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 6441 . . . 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 6435 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
6 eleq2 2314 . . . . . . . . . . . . 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 2615 . . . . 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 2628 . . 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 2510    C_ wss 3078   (/)c0 3362   Oncon0 4285  (class class class)co 5710    +o coa 6362
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-recs 6274  df-rdg 6309  df-oadd 6369
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