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Theorem int0el 4081
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 4065 . 2  |-  ( (/)  e.  A  ->  |^| A  C_  (/) )
2 0ss 3656 . . 3  |-  (/)  C_  |^| A
32a1i 11 . 2  |-  ( (/)  e.  A  ->  (/)  C_  |^| A
)
41, 3eqssd 3365 1  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3320   (/)c0 3628   |^|cint 4050
This theorem is referenced by:  intv  4375  inton  4638  onint0  4776  oev2  6767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-int 4051
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