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Theorem opthreg 6585
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6572 (via the preleq 6584 step). See df-op 3217 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207.
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 3294 . . . 4
3 preleq.3 . . . . 5
43prid1 3294 . . . 4
5 prex 3698 . . . . 5
6 prex 3698 . . . . 5
71, 5, 3, 6preleq 6584 . . . 4
82, 4, 7mpanl12 658 . . 3
9 preq1 3269 . . . . . 6
109eqeq1d 2038 . . . . 5
11 preleq.2 . . . . . 6
12 preleq.4 . . . . . 6
1311, 12preqr2 3339 . . . . 5
1410, 13syl6bi 217 . . . 4
1514imdistani 667 . . 3
168, 15syl 15 . 2
17 preq1 3269 . . . 4
1817adantr 444 . . 3
19 preq12 3271 . . . 4
2019preq2d 3276 . . 3
2118, 20eqtrd 2062 . 2
2216, 21impbii 178 1
Colors of variables: wff set class
Syntax hints:   wb 174   wa 357   wceq 1526   wcel 1528  cvv 2440  cpr 3209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1443  ax-6 1444  ax-7 1445  ax-gen 1446  ax-8 1530  ax-10 1531  ax-11 1532  ax-12 1533  ax-14 1535  ax-17 1542  ax-9 1557  ax-4 1563  ax-16 1741  ax-ext 2012  ax-sep 3626  ax-nul 3635  ax-pr 3695  ax-reg 6572
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3an 901  df-ex 1448  df-sb 1703  df-eu 1930  df-mo 1931  df-clab 2018  df-cleq 2023  df-clel 2026  df-ne 2149  df-ral 2243  df-rex 2244  df-rab 2246  df-v 2442  df-dif 2753  df-un 2755  df-in 2757  df-ss 2761  df-nul 3026  df-if 3135  df-sn 3214  df-pr 3215  df-op 3217  df-br 3532  df-opab 3585  df-eprel 3783  df-fr 3826
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