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Theorem opthpr 4181
 Description: An unordered pair has the ordered pair property (compare opth 4696) under certain conditions. (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 4179 . 2
6 idd 25 . . . 4
7 df-ne 2627 . . . . . 6
8 pm2.21 111 . . . . . 6
97, 8sylbi 198 . . . . 5
109impd 432 . . . 4
116, 10jaod 381 . . 3
12 orc 386 . . 3
1311, 12impbid1 206 . 2
145, 13syl5bb 260 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wo 369   wa 370   wceq 1437   wcel 1870   wne 2625  cvv 3087  cpr 4004 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-un 3447  df-sn 4003  df-pr 4005 This theorem is referenced by:  brdom7disj  8957  brdom6disj  8958
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