Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  elndif GIF version

Theorem elndif 3068
 Description: A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
Assertion
Ref Expression
elndif (𝐴𝐵 → ¬ 𝐴 ∈ (𝐶𝐵))

Proof of Theorem elndif
StepHypRef Expression
1 eldifn 3067 . 2 (𝐴 ∈ (𝐶𝐵) → ¬ 𝐴𝐵)
21con2i 557 1 (𝐴𝐵 → ¬ 𝐴 ∈ (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1393   ∖ cdif 2914 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920 This theorem is referenced by:  ddifnel  3075  inssdif  3173
 Copyright terms: Public domain W3C validator