Theorem neldifpr2 30292

Description: The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)

Assertion

Ref	Expression
neldifpr2	⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})

Proof of Theorem neldifpr2

Step	Hyp	Ref	Expression
1		neirr 3024	. 2 ⊢ ¬ 𝐵 ≠ 𝐵
2		eldifpr 4590	. . 3 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) ↔ (𝐵 ∈ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐵 ≠ 𝐵))
3	2	simp3bi 1142	. 2 ⊢ (𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵}) → 𝐵 ≠ 𝐵)
4	1, 3	mto 199	1 ⊢ ¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   ∈ wcel 2113   ≠ wne 3015   ∖ cdif 3926  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-v 3493  df-dif 3932  df-un 3934  df-sn 4561  df-pr 4563

This theorem is referenced by: (None)