Theorem snssl

Description: If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss . The proof of this theorem was automatically generated from snsslVD using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	snssl.1	⊢ 𝐴 ∈ V
Assertion	snssl	⊢ ( { 𝐴 } ⊆ 𝐵 → 𝐴 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		snssl.1	⊢ 𝐴 ∈ V
2	1	snid	⊢ 𝐴 ∈ { 𝐴 }
3		ssel2	⊢ ( ( { 𝐴 } ⊆ 𝐵 ∧ 𝐴 ∈ { 𝐴 } ) → 𝐴 ∈ 𝐵 )
4	2 3	mpan2	⊢ ( { 𝐴 } ⊆ 𝐵 → 𝐴 ∈ 𝐵 )