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Theorem abf 3254
 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 574 . . 3 (φx ∅)
32abssi 3009 . 2 {xφ} ⊆ ∅
4 ss0 3251 . 2 ({xφ} ⊆ ∅ → {xφ} = ∅)
53, 4ax-mp 7 1 {xφ} = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1242   ∈ wcel 1390  {cab 2023   ⊆ wss 2911  ∅c0 3218 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219 This theorem is referenced by:  csbprc  3256  mpt20  5516
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