MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preqr1 Unicode version

Theorem preqr1 4163
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
Hypotheses
Ref Expression
preqr1.1
preqr1.2
Assertion
Ref Expression
preqr1

Proof of Theorem preqr1
StepHypRef Expression
1 preqr1.1 . . . . 5
21prid1 4100 . . . 4
3 eleq2 2527 . . . 4
42, 3mpbii 211 . . 3
51elpr 4011 . . 3
64, 5sylib 196 . 2
7 preqr1.2 . . . . 5
87prid1 4100 . . . 4
9 eleq2 2527 . . . 4
108, 9mpbiri 233 . . 3
117elpr 4011 . . 3
1210, 11sylib 196 . 2
13 eqcom 2463 . 2
14 eqeq2 2469 . 2
156, 12, 13, 14oplem1 955 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  \/wo 368  =wceq 1370  e.wcel 1758   cvv 3081  {cpr 3995
This theorem is referenced by:  preqr2  4164  opthwiener  4710  cusgrafilem2  23857  wopprc  29839  2pthfrgra  31480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3083  df-un 3447  df-sn 3994  df-pr 3996
  Copyright terms: Public domain W3C validator