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Theorem opthreg 7203
 Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7190 (via the preleq 7202 step). See df-op 3553 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 3638 . . . 4
3 preleq.3 . . . . 5
43prid1 3638 . . . 4
5 prex 4111 . . . . 5
6 prex 4111 . . . . 5
71, 5, 3, 6preleq 7202 . . . 4
82, 4, 7mpanl12 666 . . 3
9 preq1 3610 . . . . . 6
109eqeq1d 2261 . . . . 5
11 preleq.2 . . . . . 6
12 preleq.4 . . . . . 6
1311, 12preqr2 3687 . . . . 5
1410, 13syl6bi 221 . . . 4
1514imdistani 674 . . 3
168, 15syl 17 . 2
17 preq1 3610 . . . 4
19 preq12 3612 . . . 4
2019preq2d 3617 . . 3
2118, 20eqtrd 2285 . 2
2216, 21impbii 182 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1619   wcel 1621  cvv 2727  cpr 3545 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-reg 7190 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-eprel 4198  df-fr 4245
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