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Theorem oprcl 3573
 Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
oprcl

Proof of Theorem oprcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex2 2570 . 2
2 df-op 3384 . . . . . . 7
32eleq2i 2104 . . . . . 6
4 df-clab 2027 . . . . . 6
53, 4bitri 173 . . . . 5
6 3simpa 901 . . . . . 6
76sbimi 1647 . . . . 5
85, 7sylbi 114 . . . 4
9 nfv 1421 . . . . 5
109sbf 1660 . . . 4
118, 10sylib 127 . . 3
1211exlimiv 1489 . 2
131, 12syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   w3a 885  wex 1381   wcel 1393  wsb 1645  cab 2026  cvv 2557  csn 3375  cpr 3376  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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-op 3384 This theorem is referenced by:  opth1  3973  opth  3974  0nelop  3985
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