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

Theorem nelaneq 9055
 Description: A class is not an element of and equal to a class at the same time. Variant of elneq 9054 analogously to elnotel 9065 and en2lp 9061. (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 9054 . . 3 (𝐴𝐵𝐴𝐵)
2 orc 863 . . . 4 𝐴𝐵 → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
3 neneq 3020 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
43olcd 872 . . . 4 (𝐴𝐵 → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
52, 4ja 188 . . 3 ((𝐴𝐵𝐴𝐵) → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
61, 5ax-mp 5 . 2 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵)
7 ianor 977 . 2 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
86, 7mpbir 233 1 ¬ (𝐴𝐵𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1530   ∈ wcel 2107   ≠ wne 3014 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-reg 9048 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-nul 4290  df-sn 4560  df-pr 4562 This theorem is referenced by:  epinid0  9056
 Copyright terms: Public domain W3C validator