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Theorem opthreg 6056
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6043 (via the preleq 6055 step). See df-op 3083 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 3146 . . . 4
3 preleq.3 . . . . 5
43prid1 3146 . . . 4
5 prex 3526 . . . . 5
6 prex 3526 . . . . 5
71, 5, 3, 6preleq 6055 . . . 4
82, 4, 7mpanl12 662 . . 3
9 preq1 3127 . . . . . 6
109eqeq1d 1937 . . . . 5
11 preleq.2 . . . . . 6
12 preleq.4 . . . . . 6
1311, 12preqr2 3186 . . . . 5
1410, 13syl6bi 218 . . . 4
1514imdistani 671 . . 3
168, 15syl 14 . 2
17 preq1 3127 . . . 4
1817adantr 448 . . 3
19 preq12 3129 . . . 4
2019preq2d 3134 . . 3
2118, 20eqtrd 1961 . 2
2216, 21impbii 178 1
Colors of variables: wff set class
Syntax hints:   wb 174   wa 360   wceq 1425   wcel 1427  cvv 2335  cpr 3075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1342  ax-6 1343  ax-7 1344  ax-gen 1345  ax-8 1429  ax-10 1430  ax-11 1431  ax-12 1432  ax-14 1434  ax-17 1441  ax-9 1456  ax-4 1462  ax-16 1640  ax-ext 1911  ax-sep 3454  ax-nul 3463  ax-pr 3523  ax-reg 6043
This theorem depends on definitions:  df-bi 175  df-or 361  df-an 362  df-3an 914  df-ex 1347  df-sb 1602  df-eu 1829  df-mo 1830  df-clab 1917  df-cleq 1922  df-clel 1925  df-ne 2049  df-ral 2142  df-rex 2143  df-v 2337  df-dif 2637  df-un 2639  df-in 2641  df-ss 2643  df-nul 2900  df-sn 3080  df-pr 3081  df-op 3083  df-br 3359  df-opab 3413  df-eprel 3608  df-fr 3649
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