Theorem elneq 9503

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 9502	. 2 ⊢ ¬ 𝐵 ∈ 𝐵
2		nelelne 3029	. 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 2113   ≠ wne 2930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-pr 5375  ax-reg 9495

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931

This theorem is referenced by:  nelaneqOLD  9505  preleqg  9522  dfac2b  10039  disjressuc2  38535  oaomoencom  43501  oenassex  43502  tfsconcat0b  43530