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Theorem in0 3321
Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
in0  |-  ( A  i^i  (/) )  =  (/)

Proof of Theorem in0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noel 3291 . . . 4  |-  -.  x  e.  (/)
21bianfi 894 . . 3  |-  ( x  e.  (/)  <->  ( x  e.  A  /\  x  e.  (/) ) )
32bicomi 131 . 2  |-  ( ( x  e.  A  /\  x  e.  (/) )  <->  x  e.  (/) )
43ineqri 3194 1  |-  ( A  i^i  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1290    e. wcel 1439    i^i cin 2999   (/)c0 3287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-in 3006  df-nul 3288
This theorem is referenced by:  0in  3322  res0  4730
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