Theorem eldifvsn 4326

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 3212	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	biantrurd 529	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵)))
3		eldifsn 4317	. 2 ⊢ (𝐴 ∈ (V ∖ {𝐵}) ↔ (𝐴 ∈ V ∧ 𝐴 ≠ 𝐵))
4	2, 3	syl6rbbr 279	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴 ≠ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by:  cnvimadfsn  7304