Theorem abf 3537

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 651	. . 3 |- ( ph -> x e. (/) )
3	2	abssi 3301	. 2 |- { x | ph } C_ (/)
4		ss0 3534	. 2 |- ( { x | ph } C_ (/) -> { x | ph } = (/) )
5	3, 4	ax-mp 5	1 |- { x | ph } = (/)

Syntax hints:   -. wn 3   = wceq 1397   e. wcel 2201   {cab 2216   C_ wss 3199   (/)c0 3493

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-dif 3201  df-in 3205  df-ss 3212  df-nul 3494

This theorem is referenced by:  csbprc  3539  mpo0  6096  fi0  7179