MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elnelne2 Structured version   Visualization version   GIF version

Theorem elnelne2 3055
Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
Assertion
Ref Expression
elnelne2 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)

Proof of Theorem elnelne2
StepHypRef Expression
1 df-nel 3044 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 3037 . 2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
31, 2sylan2b 592 1 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wcel 2098  wne 2937  wnel 3043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2720  df-clel 2806  df-ne 2938  df-nel 3044
This theorem is referenced by:  nelrnfvne  7092  eldmrexrnb  7107  absprodnn  16596  frgrncvvdeqlem2  30130  frgrncvvdeqlem3  30131  afv0nbfvbi  46560  uniimaelsetpreimafv  46765  imasetpreimafvbijlemfv1  46772  2zrngnmlid  47395  2zrngnmrid  47396
  Copyright terms: Public domain W3C validator