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Theorem abf 3501
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 123 . . 3  |-  ( ph  ->  x  e.  (/) )
32abssi 3261 . 2  |-  { x  |  ph }  C_  (/)
4 ss0 3498 . 2  |-  ( { x  |  ph }  C_  (/)  ->  { x  | 
ph }  =  (/) )
53, 4ax-mp 8 1  |-  { x  |  ph }  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   {cab 2282    C_ wss 3165   (/)c0 3468
This theorem is referenced by:  mpt20  6215  fi0  7189  pmapglb2xN  30583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469
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