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

Step	Hyp	Ref	Expression
1		abf.1	. . . 4 |- -. ph
2	1	pm2.21i 131	. . 3 |- ( ph -> x e. (/) )
3	2	abssi 3522	. 2 |- { x | ph } C_ (/)
4		ss0 3763	. 2 |- ( { x | ph } C_ (/) -> { x | ph } = (/) )
5	3, 4	ax-mp 5	1 |- { x | ph } = (/)

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