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Theorem in0 4392
Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 21-Jun-1993.)
Assertion
Ref Expression
in0 (𝐴 ∩ ∅) = ∅

Proof of Theorem in0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 noel 4331 . . . 4 ¬ 𝑥 ∈ ∅
21bianfi 535 . . 3 (𝑥 ∈ ∅ ↔ (𝑥𝐴𝑥 ∈ ∅))
32bicomi 223 . 2 ((𝑥𝐴𝑥 ∈ ∅) ↔ 𝑥 ∈ ∅)
43ineqri 4205 1 (𝐴 ∩ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  cin 3948  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-nul 4324
This theorem is referenced by:  0in  4394  csbin  4440  res0  5986  dfpo2  6296  predprc  6340  fresaun  6763  oev2  8523  dju0en  10170  ackbij1lem13  10227  ackbij1lem16  10230  incexclem  15782  bitsinv1  16383  bitsinvp1  16390  sadcadd  16399  sadadd2  16401  sadid1  16409  bitsres  16414  smumullem  16433  ressbas  17179  ressbasOLD  17180  sylow2a  19487  ablfac1eu  19943  indistopon  22504  fctop  22507  cctop  22509  rest0  22673  filconn  23387  volinun  25063  itg2cnlem2  25280  pthdlem2  29025  0pth  29378  1pthdlem2  29389  disjdifprg  31806  disjun0  31826  ofpreima2  31891  ldgenpisyslem1  33161  0elcarsg  33306  carsgclctunlem1  33316  carsgclctunlem3  33319  ballotlemfval0  33494  sate0  34406  elima4  34747  bj-rest10  35969  bj-rest0  35974  mblfinlem2  36526  conrel1d  42414  conrel2d  42415  ntrk0kbimka  42790  clsneibex  42853  neicvgbex  42863  qinioo  44248  nnfoctbdjlem  45171  caragen0  45222
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