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Theorem opnzi 3946
 Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3945). (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opth1.1 A V
opth1.2 B V
Assertion
Ref Expression
opnzi A, B⟩ ≠ ∅

Proof of Theorem opnzi
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . 3 A V
2 opth1.2 . . 3 B V
3 opm 3945 . . 3 (x x A, B⟩ ↔ (A V B V))
41, 2, 3mpbir2an 837 . 2 x x A, B
5 n0r 3211 . 2 (x x A, B⟩ → ⟨A, B⟩ ≠ ∅)
64, 5ax-mp 7 1 A, B⟩ ≠ ∅
 Colors of variables: wff set class Syntax hints:  ∃wex 1362   ∈ wcel 1374   ≠ wne 2186  Vcvv 2535  ∅c0 3201  ⟨cop 3353 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-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359 This theorem is referenced by:  0nelxp  4299  0neqopab  5473
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