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Theorem opabf 37741
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 1790 . 2 𝑥𝑦 ¬ 𝜑
3 opab0 5545 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)
42, 3mpbir 230 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1531   = wceq 1533  c0 4315  {copab 5201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-opab 5202
This theorem is referenced by:  coss0  37853
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