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Theorem opnzi 5071
Description: An ordered pair is nonempty if the arguments are sets. (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
StepHypRef Expression
1 opth1.1 . 2 𝐴 ∈ V
2 opth1.2 . 2 𝐵 ∈ V
3 opnz 5070 . 2 (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 690 1 𝐴, 𝐵⟩ ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  wne 2943  Vcvv 3351  c0 4063  cop 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324
This theorem is referenced by:  opelopabsb  5119  0nelxp  5283  unixp0  5812  funopsn  6559  0neqopab  6849  cnfldfunALT  19974  finxpreclem2  33563  finxp0  33564  finxpreclem6  33569
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