Theorem abf 3504

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 3268	. 2 |- { x | ph } C_ (/)
4		ss0 3501	. 2 |- ( { x | ph } C_ (/) -> { x | ph } = (/) )
5	3, 4	ax-mp 5	1 |- { x | ph } = (/)

Syntax hints:   -. wn 3   = wceq 1373   e. wcel 2176   {cab 2191   C_ wss 3166   (/)c0 3460

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  csbprc  3506  mpo0  6015  fi0  7077