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Theorem oaordex 6510
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 4392 . . . . 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 6509 . . . 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 6503 . . . . . . . . . . . . 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 2626 . . . . 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 2639 . . 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 3113   (/)c0 3416   Oncon0 4350  (class class class)co 5778    +o coa 6430
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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