MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abf Structured version   Unicode version

Theorem abf 3766
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 3522 . 2  |-  { x  |  ph }  C_  (/)
4 ss0 3763 . 2  |-  ( { x  |  ph }  C_  (/)  ->  { x  | 
ph }  =  (/) )
53, 4ax-mp 5 1  |-  { x  |  ph }  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   {cab 2436    C_ wss 3423   (/)c0 3732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-v 3067  df-dif 3426  df-in 3430  df-ss 3437  df-nul 3733
This theorem is referenced by:  csbprc  3768  mpt20  6252  fi0  7768  meet0  15406  join0  15407  rusgra0edg  30708  pmapglb2xN  33719
  Copyright terms: Public domain W3C validator