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Theorem abf 3406
 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 635 . . 3 (𝜑𝑥 ∈ ∅)
32abssi 3172 . 2 {𝑥𝜑} ⊆ ∅
4 ss0 3403 . 2 ({𝑥𝜑} ⊆ ∅ → {𝑥𝜑} = ∅)
53, 4ax-mp 5 1 {𝑥𝜑} = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1331   ∈ wcel 1480  {cab 2125   ⊆ wss 3071  ∅c0 3363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364 This theorem is referenced by:  csbprc  3408  mpo0  5841  fi0  6863
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