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Theorem eldifn 3167
Description: Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
Assertion
Ref Expression
eldifn (𝐴 ∈ (𝐵𝐶) → ¬ 𝐴𝐶)

Proof of Theorem eldifn
StepHypRef Expression
1 eldif 3048 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
21simprbi 271 1 (𝐴 ∈ (𝐵𝐶) → ¬ 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1463  cdif 3036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041
This theorem is referenced by:  elndif  3168  unssin  3283  inssun  3284  noel  3335  disjel  3385  undifexmid  4085  exmidundif  4097  exmidundifim  4098  phpm  6725  undifdcss  6777  fsum3cvg  11097  summodclem2a  11101  fisumss  11112  isumss2  11113  binomlem  11203  exmid1stab  13029
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