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Theorem eldifi 3164
Description: Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
Assertion
Ref Expression
eldifi (𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Proof of Theorem eldifi
StepHypRef Expression
1 eldif 3046 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
21simplbi 270 1 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1463  cdif 3034
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 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 2244  df-v 2659  df-dif 3039
This theorem is referenced by:  difss  3168  ssddif  3276  noel  3333  phpm  6712  fidifsnen  6717  elfi2  6812  fiuni  6818  fifo  6820  fzdifsuc  9754  modfzo0difsn  10061  fsum3cvg  11038  summodclem2a  11042  fisumss  11053  fsumlessfi  11121  binomlem  11144  oddprmge3  11661
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