Theorem wl-isseteq 37452

Description: A class equal to a set variable implies it is a set. Note that 𝐴 may be dependent on 𝑥. The consequent, resembling ax6ev 1968, is the accepted expression for the idea of a class being a set. Sometimes a simpler expression like the antecedent here, or in elisset 2815, is already sufficient to mark a class variable as a set. (Contributed by Wolf Lammen, 7-Sep-2025.)

Assertion

Ref	Expression
wl-isseteq	⊢ (𝑥 = 𝐴 → ∃𝑦 𝑦 = 𝐴)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem wl-isseteq

Step	Hyp	Ref	Expression
1		ax6ev 1968	. 2 ⊢ ∃𝑦 𝑦 = 𝑥
2		eqeq2 2746	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝑥 ↔ 𝑦 = 𝐴))
3	2	biimpd 229	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝑥 → 𝑦 = 𝐴))
4	3	eximdv 1916	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑦 𝑦 = 𝑥 → ∃𝑦 𝑦 = 𝐴))
5	1, 4	mpi 20	1 ⊢ (𝑥 = 𝐴 → ∃𝑦 𝑦 = 𝐴)

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2726

This theorem is referenced by: (None)