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Theorem int0el 3718
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 3703 . 2  |-  ( (/)  e.  A  ->  |^| A  C_  (/) )
2 0ss 3321 . . 3  |-  (/)  C_  |^| A
32a1i 9 . 2  |-  ( (/)  e.  A  ->  (/)  C_  |^| A
)
41, 3eqssd 3042 1  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438    C_ wss 2999   (/)c0 3286   |^|cint 3688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287  df-int 3689
This theorem is referenced by:  intv  4005  inton  4220
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