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Theorem opthreg 7319
 Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7306 (via the preleq 7318 step). See df-op 3649 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
Hypotheses
Ref Expression
preleq.1
preleq.2
preleq.3
preleq.4
Assertion
Ref Expression
opthreg

Proof of Theorem opthreg
StepHypRef Expression
1 preleq.1 . . . . 5
21prid1 3734 . . . 4
3 preleq.3 . . . . 5
43prid1 3734 . . . 4
5 prex 4217 . . . . 5
6 prex 4217 . . . . 5
71, 5, 3, 6preleq 7318 . . . 4
82, 4, 7mpanl12 663 . . 3
9 preq1 3706 . . . . . 6
109eqeq1d 2291 . . . . 5
11 preleq.2 . . . . . 6
12 preleq.4 . . . . . 6
1311, 12preqr2 3787 . . . . 5
1410, 13syl6bi 219 . . . 4
1514imdistani 671 . . 3
168, 15syl 15 . 2
17 preq1 3706 . . . 4
1817adantr 451 . . 3
19 preq12 3708 . . . 4
2019preq2d 3713 . . 3
2118, 20eqtrd 2315 . 2
2216, 21impbii 180 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   wceq 1623   wcel 1684  cvv 2788  cpr 3641 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352
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