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Theorem preleq 4306
 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 en2lp 4305 . . . . 5
2 eleq12 2144 . . . . . 6
32anbi1d 453 . . . . 5
41, 3mtbiri 633 . . . 4
54con2i 590 . . 3
65adantr 270 . 2
7 preleq.1 . . . . 5
8 preleq.2 . . . . 5
9 preleq.3 . . . . 5
10 preleq.4 . . . . 5
117, 8, 9, 10preq12b 3570 . . . 4
1211biimpi 118 . . 3
1312adantl 271 . 2
146, 13ecased 1281 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102   wo 662   wceq 1285   wcel 1434  cvv 2602  cpr 3407 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4288 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-sn 3412  df-pr 3413 This theorem is referenced by:  opthreg  4307
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