Theorem int0el 2565

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 2552	. 2 |- ((/) e. A -> |^|A (_ (/))
2		0ss 2305	. . 3 |- (/) (_ |^|A
3	2	a1i 8	. 2 |- ((/) e. A -> (/) (_ |^|A)
4	1, 3	eqssd 2082	1 |- ((/) e. A -> |^|A = (/))

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960   (_ wss 2050  (/)c0 2283  |^|cint 2537

This theorem is referenced by:  intv 2747  onint0 3013  inton 3032  oev2 4168

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-int 2538

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