Theorem int0el 3248

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 3231	. 2 |- ((/) e. A -> |^|A C_ (/))
2		0ss 2900	. . 3 |- (/) C_ |^|A
3	2	a1i 8	. 2 |- ((/) e. A -> (/) C_ |^|A)
4	1, 3	eqssd 2633	1 |- ((/) e. A -> |^|A = (/))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)c0 2875  |^|cint 3214

This theorem is referenced by:  intv 3479  inton 3720  onint0 3877  oev2 5207  inttar1 15254

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-int 3215

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