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Metamath Proof Explorer

Theorem snssg

Description: The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of Quine p. 49. (Contributed by NM, 22-Jul-2001) (Proof shortened by BJ, 1-Jan-2025)

Ref Expression
Assertion snssg ⊢ A ∈ V → A ∈ B ↔ A ⊆ B

Proof

Step Hyp Ref Expression
1 snssb ⊢ A ⊆ B ↔ A ∈ V → A ∈ B
2 1 bicomi ⊢ A ∈ V → A ∈ B ↔ A ⊆ B
3 elex ⊢ A ∈ V → A ∈ V
4 imbibi ⊢ A ∈ V → A ∈ B ↔ A ⊆ B → A ∈ V → A ∈ B ↔ A ⊆ B
5 2 3 4 mpsyl ⊢ A ∈ V → A ∈ B ↔ A ⊆ B