Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opabf Structured version   Visualization version   GIF version

Theorem opabf 38350
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 1793 . 2 𝑥𝑦 ¬ 𝜑
3 opab0 5564 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)
42, 3mpbir 231 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1535   = wceq 1537  c0 4339  {copab 5210
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211
This theorem is referenced by:  coss0  38461
  Copyright terms: Public domain W3C validator