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Theorem opthreg 6132
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6119 (via the preleq 6131 step). See df-op 3082 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 3145 . . . 4
3 preleq.3 . . . . 5
43prid1 3145 . . . 4
5 prex 3526 . . . . 5
6 prex 3526 . . . . 5
71, 5, 3, 6preleq 6131 . . . 4
82, 4, 7mpanl12 657 . . 3
9 preq1 3126 . . . . . 6
109eqeq1d 1925 . . . . 5
11 preleq.2 . . . . . 6
12 preleq.4 . . . . . 6
1311, 12preqr2 3186 . . . . 5
1410, 13syl6bi 216 . . . 4
1514imdistani 666 . . 3
168, 15syl 14 . 2
17 preq1 3126 . . . 4
1817adantr 443 . . 3
19 preq12 3128 . . . 4
2019preq2d 3133 . . 3
2118, 20eqtrd 1949 . 2
2216, 21impbii 177 1
Colors of variables: wff set class
Syntax hints:   wb 173   wa 356   wceq 1413   wcel 1415  cvv 2322  cpr 3074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1330  ax-6 1331  ax-7 1332  ax-gen 1333  ax-8 1417  ax-10 1418  ax-11 1419  ax-12 1420  ax-14 1422  ax-17 1429  ax-9 1444  ax-4 1450  ax-16 1628  ax-ext 1899  ax-sep 3454  ax-nul 3463  ax-pr 3523  ax-reg 6119
This theorem depends on definitions:  df-bi 174  df-or 357  df-an 358  df-3an 900  df-ex 1335  df-sb 1590  df-eu 1817  df-mo 1818  df-clab 1905  df-cleq 1910  df-clel 1913  df-ne 2036  df-ral 2129  df-rex 2130  df-v 2324  df-dif 2635  df-un 2637  df-in 2639  df-ss 2641  df-nul 2899  df-sn 3079  df-pr 3080  df-op 3082  df-br 3359  df-opab 3413  df-eprel 3608  df-fr 3650
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