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Theorem opthreg 4471
 Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4452 (via the preleq 4470 step). See df-op 3536 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 3629 . . . 4
3 preleq.3 . . . . 5
43prid1 3629 . . . 4
5 preleq.2 . . . . . 6
6 prexg 4133 . . . . . 6
71, 5, 6mp2an 422 . . . . 5
8 preleq.4 . . . . . 6
9 prexg 4133 . . . . . 6
103, 8, 9mp2an 422 . . . . 5
111, 7, 3, 10preleq 4470 . . . 4
122, 4, 11mpanl12 432 . . 3
13 preq1 3600 . . . . . 6
1413eqeq1d 2148 . . . . 5
155, 8preqr2 3696 . . . . 5
1614, 15syl6bi 162 . . . 4
1716imdistani 441 . . 3
1812, 17syl 14 . 2
19 preq1 3600 . . . 4
21 preq12 3602 . . . 4
2221preq2d 3607 . . 3
2320, 22eqtrd 2172 . 2
2418, 23impbii 125 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104   wceq 1331   wcel 1480  cvv 2686  cpr 3528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pr 4131  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-sn 3533  df-pr 3534 This theorem is referenced by: (None)
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