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Theorem preleq 8122
 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 4151 . . . . . 6
65biimpi 198 . . . . 5
76ord 379 . . . 4
8 en2lp 8118 . . . . 5
9 eleq12 2519 . . . . . 6
109anbi1d 711 . . . . 5
118, 10mtbiri 305 . . . 4
127, 11syl6 34 . . 3
1312con4d 109 . 2
1413impcom 432 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 370   wa 371   wceq 1444   wcel 1887  cvv 3045  cpr 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-reg 8107 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-eprel 4745  df-fr 4793 This theorem is referenced by:  opthreg  8123  dfac2  8561
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