Theorem bj-df-clel 32081

Description: Candidate definition for df-clel 2606 (the need for it is exposed in bj-ax8 32080). The similarity of the hypothesis and the conclusion, together with all possible dv conditions, makes it clear that this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This definition should be directly referenced only by bj-dfclel 32082, which should be used instead. The proof is irrelevant since this is a proposal for an axiom.

Note: the current definition df-clel 2606 already mentions cleljust 1985 as a justification; here, we merely propose to put it as a hypothesis to make things clearer. (Contributed by BJ, 27-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-df-clel.1	⊢ (𝑢 ∈ 𝑣 ↔ ∃𝑤(𝑤 = 𝑢 ∧ 𝑤 ∈ 𝑣))

Assertion

Ref	Expression
bj-df-clel	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑣,𝑢,𝑤,𝑥,𝐴   𝑢,𝐵,𝑣,𝑤,𝑥

Proof of Theorem bj-df-clel

Step	Hyp	Ref	Expression
1		df-clel 2606	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977

This theorem depends on definitions:  df-clel 2606

This theorem is referenced by:  bj-dfclel  32082