Theorem opabf 38324

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 1794	. 2 ⊢ ∀𝑥∀𝑦 ¬ 𝜑
3		opab0 5573	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)
4	2, 3	mpbir 231	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3  ∀wal 1535   = wceq 1537  ∅c0 4352  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229

This theorem is referenced by:  coss0  38435