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Theorem abf 3448
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 636 . . 3  |-  ( ph  ->  x  e.  (/) )
32abssi 3213 . 2  |-  { x  |  ph }  C_  (/)
4 ss0 3445 . 2  |-  ( { x  |  ph }  C_  (/)  ->  { x  | 
ph }  =  (/) )
53, 4ax-mp 5 1  |-  { x  |  ph }  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1342    e. wcel 2135   {cab 2150    C_ wss 3112   (/)c0 3405
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-dif 3114  df-in 3118  df-ss 3125  df-nul 3406
This theorem is referenced by:  csbprc  3450  mpo0  5904  fi0  6932
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