Theorem abf 3512

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 647	. . 3 |- ( ph -> x e. (/) )
3	2	abssi 3276	. 2 |- { x | ph } C_ (/)
4		ss0 3509	. 2 |- ( { x | ph } C_ (/) -> { x | ph } = (/) )
5	3, 4	ax-mp 5	1 |- { x | ph } = (/)

Syntax hints:   -. wn 3   = wceq 1373   e. wcel 2178   {cab 2193   C_ wss 3174   (/)c0 3468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by:  csbprc  3514  mpo0  6038  fi0  7103