HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem eldifn 2159
Description: Implication of membership in a class difference.
Assertion
Ref Expression
eldifn |- (A e. (B \ C) -> -. A e. C)
Proof of Theorem eldifn
StepHypRef Expression
1 eldif 2053 . 2 |- (A e. (B \ C) <-> (A e. B /\ -. A e. C))
21pm3.27bi 326 1 |- (A e. (B \ C) -> -. A e. C)
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   e. wcel 956   \ cdif 2040
This theorem is referenced by:  elndif 2160  tz7.7 2968  tfi 3121  peano5 3148  tz7.48-2 3948  tz7.49 3950  inf3lem3 4595  setind 4628  acdc3lem 7436  acdc2lem1 7438  acdclem 7444  clsval2 7635  elcls 7654  bcthlem28 7976  strlem1 10115
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-dif 2045
Copyright terms: Public domain