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Theorem in0 3493
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 3472 . . . 4  |-  -.  x  e.  (/)
21bianfi 891 . . 3  |-  ( x  e.  (/)  <->  ( x  e.  A  /\  x  e.  (/) ) )
32bicomi 193 . 2  |-  ( ( x  e.  A  /\  x  e.  (/) )  <->  x  e.  (/) )
43ineqri 3375 1  |-  ( A  i^i  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164   (/)c0 3468
This theorem is referenced by:  res0  4975  fresaun  5428  fnsuppeq0  5749  oev2  6538  cda0en  7821  ackbij1lem13  7874  ackbij1lem16  7877  incexclem  12311  bitsinv1  12649  bitsinvp1  12656  sadcadd  12665  sadadd2  12667  sadid1  12675  bitsres  12680  smumullem  12699  ressbas  13214  sylow2a  14946  ablfac1eu  15324  indistopon  16754  fctop  16757  cctop  16759  rest0  16916  restsn  16917  filcon  17594  volinun  18919  itg2cnlem2  19133  chtdif  20412  ppidif  20417  ppi1  20418  cht1  20419  ballotlemfp1  23066  ballotlemfval0  23070  ballotlemgun  23099  disjdifprg  23367  measvuni  23557  measinb  23563  cndprobnul  23655  dfpo2  24183  pred0  24270  neiopne  25154  hdrmp  25809  0pth  28356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-in 3172  df-nul 3469
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