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Theorem opth1 4725
Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opth1.1
opth1.2
Assertion
Ref Expression
opth1

Proof of Theorem opth1
StepHypRef Expression
1 opth1.1 . . . 4
21sneqr 4197 . . 3
32a1i 11 . 2
4 opth1.2 . . . . . . . . 9
51, 4opi1 4719 . . . . . . . 8
6 id 22 . . . . . . . 8
75, 6syl5eleq 2551 . . . . . . 7
8 oprcl 4242 . . . . . . 7
97, 8syl 16 . . . . . 6
109simpld 459 . . . . 5
11 prid1g 4136 . . . . 5
1210, 11syl 16 . . . 4
13 eleq2 2530 . . . 4
1412, 13syl5ibrcom 222 . . 3
15 elsni 4054 . . . 4
1615eqcomd 2465 . . 3
1714, 16syl6 33 . 2
18 dfopg 4215 . . . . 5
197, 8, 183syl 20 . . . 4
207, 19eleqtrd 2547 . . 3
21 elpri 4049 . . 3
2220, 21syl 16 . 2
233, 17, 22mpjaod 381 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  {csn 4029  {cpr 4031  <.cop 4035
This theorem is referenced by:  opth  4726  dmsnopg  5484  funcnvsn  5638  oprabid  6323  seqomlem2  7135  unxpdomlem3  7746  dfac5lem4  8528  dcomex  8848  canthwelem  9049  uzrdgfni  12069  gsum2d2  17002  2trllemA  24552  2pthon  24604  2pthon3v  24606  constr3lem2  24646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036
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