Theorem eldifvsn 4732

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		elex 3514	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	biantrurd 535	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵)))
3		eldifsn 4721	. 2 ⊢ (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵))
4	2, 3	syl6rbbr 292	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2114   ≠ wne 3018  Vcvv 3496   ∖ cdif 3935  {csn 4569

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-v 3498  df-dif 3941  df-sn 4570

This theorem is referenced by:  cnvimadfsn  7841