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Metamath Proof Explorer

Theorem issetft

Description: Closed theorem form of isset that does not require x and A to be distinct. Extracted from the proof of vtoclgft . (Contributed by Wolf Lammen, 9-Apr-2025)

Ref Expression
Assertion issetft ⊢ Ⅎ _ x A → A ∈ V ↔ ∃ x x = A

Proof

Step Hyp Ref Expression
1 isset ⊢ A ∈ V ↔ ∃ y y = A
2 cbvexeqsetf ⊢ Ⅎ _ x A → ∃ x x = A ↔ ∃ y y = A
3 1 2 bitr4id ⊢ Ⅎ _ x A → A ∈ V ↔ ∃ x x = A