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Theorem in0 3422
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
StepHypRef Expression
1 noel 3401 . . . 4  |-  -.  x  e.  (/)
21bianfi 896 . . 3  |-  ( x  e.  (/)  <->  ( x  e.  A  /\  x  e.  (/) ) )
32bicomi 195 . 2  |-  ( ( x  e.  A  /\  x  e.  (/) )  <->  x  e.  (/) )
43ineqri 3304 1  |-  ( A  i^i  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1619    e. wcel 1621    i^i cin 3093   (/)c0 3397
This theorem is referenced by:  res0  4912  fresaun  5315  fnsuppeq0  5632  oev2  6455  cda0en  7738  ackbij1lem13  7791  ackbij1lem16  7794  bitsinv1  12560  bitsinvp1  12567  sadcadd  12576  sadadd2  12578  sadid1  12586  bitsres  12591  smumullem  12610  ressbas  13125  sylow2a  14857  ablfac1eu  15235  indistopon  16665  fctop  16668  cctop  16670  rest0  16827  restsn  16828  filcon  17505  volinun  18830  itg2cnlem2  19044  chtdif  20323  ppidif  20328  ppi1  20329  cht1  20330  ballotlemfp1  22976  ballotlemfval0  22980  ballotlemgun  23009  dfpo2  23448  pred0  23533  neiopne  24382  hdrmp  25038
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-dif 3097  df-in 3101  df-nul 3398
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