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Theorem opnzi 4972
 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 4971 . 2 (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 975 1 𝐴, 𝐵⟩ ≠ ∅
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2030   ≠ wne 2823  Vcvv 3231  ∅c0 3948  ⟨cop 4216 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217 This theorem is referenced by:  opelopabsb  5014  0nelxp  5177  0nelxpOLD  5178  unixp0  5707  funopsn  6453  0neqopab  6740  cnfldfunALT  19807  finxpreclem2  33357  finxp0  33358  finxpreclem6  33363
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