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Theorem abf 3780
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 136 . . 3  |-  ( ph  ->  x  e.  (/) )
32abssi 3516 . 2  |-  { x  |  ph }  C_  (/)
4 ss0 3777 . 2  |-  ( { x  |  ph }  C_  (/)  ->  { x  | 
ph }  =  (/) )
53, 4ax-mp 5 1  |-  { x  |  ph }  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1455    e. wcel 1898   {cab 2448    C_ wss 3416   (/)c0 3743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744
This theorem is referenced by:  csbprc  3782  mpt20  6388  fi0  7960  meet0  16432  join0  16433  rusgra0edg  25732  pmapglb2xN  33382
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