Theorem abf 4359

Description: A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)

Hypothesis

Ref	Expression
abf.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
abf	⊢ {𝑥 ∣ 𝜑} = ∅

Proof of Theorem abf

Step	Hyp	Ref	Expression
1		ab0 4336	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
2		abf.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbir 1799	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1536  {cab 2802  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-dif 3942  df-nul 4295

This theorem is referenced by:  csbprc  4361  mpo0  7242  fi0  8887  meet0  17750  join0  17751  fmla0disjsuc  32649  0qs  35626  pmapglb2xN  36912