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

Theorem nelaneq 9543
Description: A class is not an element of and equal to a class at the same time. Variant of elneq 9542 analogously to elnotel 9554 and en2lp 9550. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.)
Assertion
Ref Expression
nelaneq ¬ (𝐴𝐵𝐴 = 𝐵)

Proof of Theorem nelaneq
StepHypRef Expression
1 elneq 9542 . . 3 (𝐴𝐵𝐴𝐵)
2 orc 866 . . . 4 𝐴𝐵 → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
3 neneq 2946 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
43olcd 873 . . . 4 (𝐴𝐵 → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
52, 4ja 186 . . 3 ((𝐴𝐵𝐴𝐵) → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
61, 5ax-mp 5 . 2 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵)
7 ianor 981 . 2 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
86, 7mpbir 230 1 ¬ (𝐴𝐵𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-pr 5388  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3449  df-un 3919  df-sn 4591  df-pr 4593
This theorem is referenced by:  epinid0  9544
  Copyright terms: Public domain W3C validator