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Theorem opabf 37840
Description: A class abstraction of a collection of ordered pairs with a negated wff is the empty set. (Contributed by Peter Mazsa, 21-Oct-2019.) (Proof shortened by Thierry Arnoux, 18-Feb-2022.)
Hypothesis
Ref Expression
opabf.1 ¬ 𝜑
Assertion
Ref Expression
opabf {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅

Proof of Theorem opabf
StepHypRef Expression
1 opabf.1 . . 3 ¬ 𝜑
21gen2 1791 . 2 𝑥𝑦 ¬ 𝜑
3 opab0 5556 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)
42, 3mpbir 230 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1532   = wceq 1534  c0 4323  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5211
This theorem is referenced by:  coss0  37951
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