Theorem int0el 1985

Description: The intersection of a class containing the empty set is empty.

Assertion

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

Proof of Theorem int0el

Step	Hyp	Ref	Expression
1		intss1 1979	. 2 |- ((/) e. A -> |^|A (_ (/))
2		0ss 1725	. . 3 |- (/) (_ |^|A
3	2	a1i 7	. 2 |- ((/) e. A -> (/) (_ |^|A)
4	1, 3	eqssd 1518	1 |- ((/) e. A -> |^|A = (/))

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707  |^|cint 1965

This theorem is referenced by:  onint0 2262  inton 2281

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-int 1966

metamath.org