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Theorem oaordex 6730
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 4557 . . . . 5  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
21adantl 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  A  C_  B )
)
3 oawordex 6729 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  E. x  e.  On  ( A  +o  x )  =  B ) )
42, 3sylibd 206 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  ->  E. x  e.  On  ( A  +o  x
)  =  B ) )
5 oaord1 6723 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
6 eleq2 2441 . . . . . . . . . . . . 13  |-  ( ( A  +o  x )  =  B  ->  ( A  e.  ( A  +o  x )  <->  A  e.  B ) )
75, 6sylan9bb 681 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  x  e.  On )  /\  ( A  +o  x )  =  B )  ->  ( (/)  e.  x  <->  A  e.  B ) )
87biimprcd 217 . . . . . . . . . . 11  |-  ( A  e.  B  ->  (
( ( A  e.  On  /\  x  e.  On )  /\  ( A  +o  x )  =  B )  ->  (/)  e.  x
) )
98exp4c 592 . . . . . . . . . 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 574 . . . . . . . 8  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( ( x  e.  On  /\  ( A  +o  x )  =  B )  ->  (/)  e.  x
) )
12 simpr 448 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( A  +o  x
)  =  B )  ->  ( A  +o  x )  =  B )
1312a1i 11 . . . . . . . 8  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( ( x  e.  On  /\  ( A  +o  x )  =  B )  ->  ( A  +o  x )  =  B ) )
1411, 13jcad 520 . . . . . . 7  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( ( x  e.  On  /\  ( A  +o  x )  =  B )  ->  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
1514exp3a 426 . . . . . 6  |-  ( ( A  e.  On  /\  A  e.  B )  ->  ( x  e.  On  ->  ( ( A  +o  x )  =  B  ->  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
1615reximdvai 2752 . . . . 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 424 . . . 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 452 . . 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 199 . . . . . . 7  |-  ( ( ( A  e.  On  /\  x  e.  On )  /\  ( A  +o  x )  =  B )  ->  ( (/)  e.  x  ->  A  e.  B ) )
2120exp31 588 . . . . . 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 573 . . . 4  |-  ( A  e.  On  ->  (
x  e.  On  ->  ( ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) ) )
2423rexlimdv 2765 . . 3  |-  ( A  e.  On  ->  ( E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2524adantr 452 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2619, 25impbid 184 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 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2643    C_ wss 3256   (/)c0 3564   Oncon0 4515  (class class class)co 6013    +o coa 6650
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-oadd 6657
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