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Theorem abf 3303
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 608 . . 3  |-  ( ph  ->  x  e.  (/) )
32abssi 3078 . 2  |-  { x  |  ph }  C_  (/)
4 ss0 3300 . 2  |-  ( { x  |  ph }  C_  (/)  ->  { x  | 
ph }  =  (/) )
53, 4ax-mp 7 1  |-  { x  |  ph }  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1285    e. wcel 1434   {cab 2069    C_ wss 2982   (/)c0 3267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-nul 3268
This theorem is referenced by:  csbprc  3305  mpt20  5625
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