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Theorem int0el 3930
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 3914 . 2  |-  ( (/)  e.  A  ->  |^| A  C_  (/) )
2 0ss 3517 . . 3  |-  (/)  C_  |^| A
32a1i 10 . 2  |-  ( (/)  e.  A  ->  (/)  C_  |^| A
)
41, 3eqssd 3230 1  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    C_ wss 3186   (/)c0 3489   |^|cint 3899
This theorem is referenced by:  intv  4223  inton  4486  onint0  4624  oev2  6564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-dif 3189  df-in 3193  df-ss 3200  df-nul 3490  df-int 3900
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