Theorem elneq 9563

Description: A class is not equal to any of its elements. (Contributed by AV, 14-Jun-2022.)

Assertion

Ref	Expression
elneq	⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)

Proof of Theorem elneq

Step	Hyp	Ref	Expression
1		elirr 9562	. 2 ⊢ ¬ 𝐵 ∈ 𝐵
2		nelelne 3065	. 2 ⊢ (¬ 𝐵 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2149   ≠ wne 2964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-reg 9554

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965

This theorem is referenced by:  nelaneqOLDOLD  9566  preleqg  9584  dfac2b  10114  disjressuc2  38950  oaomoencom  43936  oenassex  43937  tfsconcat0b  43965