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Theorem opthreg 6195
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6182 (via the preleq 6194 step). See df-op 3088 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 3153 . . . 4
3 preleq.3 . . . . 5
43prid1 3153 . . . 4
5 prex 3538 . . . . 5
6 prex 3538 . . . . 5
71, 5, 3, 6preleq 6194 . . . 4
82, 4, 7mpanl12 658 . . 3
9 preq1 3134 . . . . . 6
109eqeq1d 1926 . . . . 5
11 preleq.2 . . . . . 6
12 preleq.4 . . . . . 6
1311, 12preqr2 3195 . . . . 5
1410, 13syl6bi 217 . . . 4
1514imdistani 667 . . 3
168, 15syl 15 . 2
17 preq1 3134 . . . 4
1817adantr 444 . . 3
19 preq12 3136 . . . 4
2019preq2d 3141 . . 3
2118, 20eqtrd 1950 . 2
2216, 21impbii 178 1
Colors of variables: wff set class
Syntax hints:   wb 174   wa 357   wceq 1414   wcel 1416  cvv 2324  cpr 3080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1331  ax-6 1332  ax-7 1333  ax-gen 1334  ax-8 1418  ax-10 1419  ax-11 1420  ax-12 1421  ax-14 1423  ax-17 1430  ax-9 1445  ax-4 1451  ax-16 1629  ax-ext 1900  ax-sep 3466  ax-nul 3475  ax-pr 3535  ax-reg 6182
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3an 901  df-ex 1336  df-sb 1591  df-eu 1818  df-mo 1819  df-clab 1906  df-cleq 1911  df-clel 1914  df-ne 2037  df-ral 2131  df-rex 2132  df-v 2326  df-dif 2637  df-un 2639  df-in 2641  df-ss 2645  df-nul 2903  df-sn 3085  df-pr 3086  df-op 3088  df-br 3371  df-opab 3425  df-eprel 3620  df-fr 3662
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