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Theorem preleq 7573
 Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
Hypotheses
Ref Expression
preleq.1
preleq.2
preleq.3
preleq.4
Assertion
Ref Expression
preleq

Proof of Theorem preleq
StepHypRef Expression
1 preleq.1 . . . . . . 7
2 preleq.2 . . . . . . 7
3 preleq.3 . . . . . . 7
4 preleq.4 . . . . . . 7
51, 2, 3, 4preq12b 3975 . . . . . 6
65biimpi 188 . . . . 5
76ord 368 . . . 4
8 en2lp 7572 . . . . 5
9 eleq12 2499 . . . . . 6
109anbi1d 687 . . . . 5
118, 10mtbiri 296 . . . 4
127, 11syl6 32 . . 3
1312con4d 100 . 2
1413impcom 421 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 359   wa 360   wceq 1653   wcel 1726  cvv 2957  cpr 3816 This theorem is referenced by:  opthreg  7574  dfac2  8012 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-reg 7561 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-eprel 4495  df-fr 4542
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