Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  abf Structured version   Visualization version   GIF version

Theorem abf 4121
 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 {𝑥𝜑} = ∅

Proof of Theorem abf
StepHypRef Expression
1 ab0 4094 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
2 abf.1 . 2 ¬ 𝜑
31, 2mpgbir 1875 1 {𝑥𝜑} = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1632  {cab 2746  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-nul 4059 This theorem is referenced by:  csbprc  4123  csbprcOLD  4124  mpt20  6890  fi0  8491  meet0  17338  join0  17339  0qs  34455  pmapglb2xN  35561
 Copyright terms: Public domain W3C validator