Theorem int0el 3979

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 3964	. 2 |- ( (/) e. A -> |^| A C_ (/) )
2		0ss 3547	. . 3 |- (/) C_ |^| A
3	2	a1i 9	. 2 |- ( (/) e. A -> (/) C_ |^| A )
4	1, 3	eqssd 3255	1 |- ( (/) e. A -> |^| A = (/) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   C_ wss 3211   (/)c0 3508   |^|cint 3949

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509  df-int 3950

This theorem is referenced by:  intv  4283  inton  4514