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Theorem int0el 3893
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
StepHypRef Expression
1 intss1 3877 . 2  |-  ( (/)  e.  A  ->  |^| A  C_  (/) )
2 0ss 3483 . . 3  |-  (/)  C_  |^| A
32a1i 10 . 2  |-  ( (/)  e.  A  ->  (/)  C_  |^| A
)
41, 3eqssd 3196 1  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   |^|cint 3862
This theorem is referenced by:  intv  4186  inton  4449  onint0  4587  oev2  6522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-int 3863
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