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Theorem moop2 3988
 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 2058 . . . 4
2 moop2.1 . . . . . 6
3 vex 2560 . . . . . 6
42, 3opth 3974 . . . . 5
54simprbi 260 . . . 4
61, 5syl 14 . . 3
76gen2 1339 . 2
8 nfcsb1v 2882 . . . . 5
9 nfcv 2178 . . . . 5
108, 9nfop 3565 . . . 4
1110nfeq2 2189 . . 3
12 csbeq1a 2860 . . . . 5
13 id 19 . . . . 5
1412, 13opeq12d 3557 . . . 4
1514eqeq2d 2051 . . 3
1611, 15mo4f 1960 . 2
177, 16mpbir 134 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97  wal 1241   wceq 1243   wcel 1393  wmo 1901  cvv 2557  csb 2852  cop 3378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384 This theorem is referenced by: (None)
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