Theorem int0el 3929

Description: The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
int0el	|- ( (/) e. A -> |^| A = (/) )

Proof of Theorem int0el

Step	Hyp	Ref	Expression
1		intss1 3914	. 2 |- ( (/) e. A -> |^| A C_ (/) )
2		0ss 3507	. . 3 |- (/) C_ |^| A
3	2	a1i 9	. 2 |- ( (/) e. A -> (/) C_ |^| A )
4	1, 3	eqssd 3218	1 |- ( (/) e. A -> |^| A = (/) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   C_ wss 3174   (/)c0 3468   |^|cint 3899

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  df-int 3900

This theorem is referenced by:  intv  4230  inton  4458