Theorem opabf 38624

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 1798	. 2 ⊢ ∀𝑥∀𝑦 ¬ 𝜑
3		opab0 5510	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)
4	2, 3	mpbir 231	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3  ∀wal 1540   = wceq 1542  ∅c0 4287  {copab 5162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163

This theorem is referenced by:  coss0  38817