Theorem opabf 36425

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

Step	Hyp	Ref	Expression
1		opabf.1	. . 3 ⊢ ¬ 𝜑
2	1	gen2 1800	. 2 ⊢ ∀𝑥∀𝑦 ¬ 𝜑
3		opab0 5460	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)
4	2, 3	mpbir 230	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3  ∀wal 1537   = wceq 1539  ∅c0 4253  {copab 5132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133

This theorem is referenced by:  coss0  36524