Theorem abf 3780

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 136	. . 3 |- ( ph -> x e. (/) )
3	2	abssi 3516	. 2 |- { x | ph } C_ (/)
4		ss0 3777	. 2 |- ( { x | ph } C_ (/) -> { x | ph } = (/) )
5	3, 4	ax-mp 5	1 |- { x | ph } = (/)

Syntax hints:   -. wn 3   = wceq 1455   e. wcel 1898   {cab 2448   C_ wss 3416   (/)c0 3743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744

This theorem is referenced by:  csbprc  3782  mpt20  6388  fi0  7960  meet0  16432  join0  16433  rusgra0edg  25732  pmapglb2xN  33382