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Theorem moop2 4751
 Description: "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
moop2.1
Assertion
Ref Expression
moop2
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem moop2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqtr2 2484 . . . 4
2 moop2.1 . . . . . 6
3 vex 3112 . . . . . 6
42, 3opth 4730 . . . . 5
54simprbi 464 . . . 4
61, 5syl 16 . . 3
76gen2 1620 . 2
8 nfcsb1v 3446 . . . . 5
9 nfcv 2619 . . . . 5
108, 9nfop 4235 . . . 4
1110nfeq2 2636 . . 3
12 csbeq1a 3439 . . . . 5
13 id 22 . . . . 5
1412, 13opeq12d 4227 . . . 4
1514eqeq2d 2471 . . 3
1611, 15mo4f 2337 . 2
177, 16mpbir 209 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369  wal 1393   wceq 1395   wcel 1819  wmo 2284  cvv 3109  csb 3430  cop 4038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039 This theorem is referenced by:  euop2  4756
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