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Theorem opabf 38370
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 1795 . 2 𝑥𝑦 ¬ 𝜑
3 opab0 5558 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)
42, 3mpbir 231 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1537   = wceq 1539  c0 4332  {copab 5204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205
This theorem is referenced by:  coss0  38481
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