Theorem elissetv 2843

Description: An element of a class exists. Version of elisset 2844 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2741. Prefer its use over elisset 2844 when sufficient (for instance in usages where 𝑥 is a dummy variable). (Contributed by BJ, 14-Sep-2019.)

Assertion

Ref	Expression
elissetv	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem elissetv

Step	Hyp	Ref	Expression
1		dfclel 2838	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		exsimpl 1897	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylbi 220	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-clel 2837

This theorem is used by:  elisset  2844  clelab  2906  isseti  3472  elex  3475  elex22  3478  spcimgft  3514  kardeq0  35577  bj-issetiv  37540  bj-ceqsaltv  37550  bj-ceqsalgv  37554  bj-spcimdvv  37559  bj-vtoclg1fv  37582  bj-vtoclg  37583  bj-ru  37608  bj-unexg  37702