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Theorem abf 3828
Description: A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
Hypothesis
Ref Expression
abf.1  |-  -.  ph
Assertion
Ref Expression
abf  |-  { x  |  ph }  =  (/)

Proof of Theorem abf
StepHypRef Expression
1 abf.1 . . . 4  |-  -.  ph
21pm2.21i 131 . . 3  |-  ( ph  ->  x  e.  (/) )
32abssi 3571 . 2  |-  { x  |  ph }  C_  (/)
4 ss0 3825 . 2  |-  ( { x  |  ph }  C_  (/)  ->  { x  | 
ph }  =  (/) )
53, 4ax-mp 5 1  |-  { x  |  ph }  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   {cab 2442    C_ wss 3471   (/)c0 3793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794
This theorem is referenced by:  csbprc  3830  mpt20  6366  fi0  7898  meet0  15894  join0  15895  rusgra0edg  25082  pmapglb2xN  35639
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