Theorem int0el 3995

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 3980	. 2 |- ( (/) e. A -> |^| A C_ (/) )
2		0ss 3561	. . 3 |- (/) C_ |^| A
3	2	a1i 9	. 2 |- ( (/) e. A -> (/) C_ |^| A )
4	1, 3	eqssd 3265	1 |- ( (/) e. A -> |^| A = (/) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520   |^|cint 3965

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-int 3966

This theorem is referenced by:  intv  4302  inton  4533