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Theorem oaordex 6793
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 4615 . . . . 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 6792 . . . 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 6786 . . . . . . . . . . . . 13  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
6 eleq2 2496 . . . . . . . . . . . . 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 2808 . . . . 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 2821 . . 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 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   (/)c0 3620   Oncon0 4573  (class class class)co 6073    +o coa 6713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-oadd 6720
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