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Theorem opthpr 3706
 Description: A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
Hypotheses
Ref Expression
preq12b.1
preq12b.2
preq12b.3
preq12b.4
Assertion
Ref Expression
opthpr

Proof of Theorem opthpr
StepHypRef Expression
1 preq12b.1 . . 3
2 preq12b.2 . . 3
3 preq12b.3 . . 3
4 preq12b.4 . . 3
51, 2, 3, 4preq12b 3704 . 2
6 idd 21 . . . 4
7 df-ne 2310 . . . . . 6
8 pm2.21 607 . . . . . 6
97, 8sylbi 120 . . . . 5
109impd 252 . . . 4
116, 10jaod 707 . . 3
12 orc 702 . . 3
1311, 12impbid1 141 . 2
145, 13syl5bb 191 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wo 698   wceq 1332   wcel 1481   wne 2309  cvv 2689  cpr 3532 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-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538 This theorem is referenced by: (None)
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