Theorem opabf 35614

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 1793	. 2 ⊢ ∀𝑥∀𝑦 ¬ 𝜑
3		opab0 5433	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)
4	2, 3	mpbir 233	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3  ∀wal 1531   = wceq 1533  ∅c0 4290  {copab 5120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5121

This theorem is referenced by:  coss0  35713