Theorem wl-dfclel 38021

Description: The defining characterization of class membership. Unlike the forms on which it is based, it is unrestricted. Proven in Tarski's FOL, from the axiom of (set) extensionality (ax-ext 2737), the definitions df-clel 2840 and df-cleq . (Contributed by BJ, 27-Jun-2019.) Base on wl-dfclel.just 38019. (Revised by Wolf Lammen, 13-Apr-2026.)

Assertion

Ref	Expression
wl-dfclel	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Distinct variable group:   𝑥,𝐴,𝐵

Proof of Theorem wl-dfclel

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2769	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
2		eleq1w 2848	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
3	1, 2	anbi12d 643	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)))
4	3	cbvexvw 2060	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵))
5	4	wl-dfclel.just 38019	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-cleq 2757  df-clel 2840

This theorem is referenced by: (None)