ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abf GIF version

Theorem abf 3466
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 abf.1 . . . 4 ¬ 𝜑
21pm2.21i 646 . . 3 (𝜑𝑥 ∈ ∅)
32abssi 3230 . 2 {𝑥𝜑} ⊆ ∅
4 ss0 3463 . 2 ({𝑥𝜑} ⊆ ∅ → {𝑥𝜑} = ∅)
53, 4ax-mp 5 1 {𝑥𝜑} = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1353  wcel 2148  {cab 2163  wss 3129  c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by:  csbprc  3468  mpo0  5942  fi0  6971
  Copyright terms: Public domain W3C validator