Theorem elneq 9632

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 9631	. 2 ⊢ ¬ 𝐵 ∈ 𝐵
2		nelelne 3060	. 2 ⊢ (¬ 𝐵 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147   ≠ wne 2959

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-reg 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960

This theorem is used by:  nelaneqOLDOLD  9635  preleqg  9653  dfac2b  10248  disjressuc2  39609  oaomoencom  44580  oenassex  44581  tfsconcat0b  44609