Theorem eldifvsn 4751

Description: A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)

Assertion

Ref	Expression
eldifvsn	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵))

Proof of Theorem eldifvsn

Step	Hyp	Ref	Expression
1		eldifsn 4740	. 2 ⊢ (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵))
2		elex 3459	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3	2	biantrurd 532	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵)))
4	1, 3	bitr4id 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ≠ wne 2930  Vcvv 3438   ∖ cdif 3896  {csn 4578

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-v 3440  df-dif 3902  df-sn 4579

This theorem is referenced by:  cnvimadfsn  8112