Theorem snssl 42450

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 4719. The proof of this theorem was automatically generated from snsslVD 42449 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.)

Hypothesis

Ref	Expression
snssl.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snssl	⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Proof of Theorem snssl

Step	Hyp	Ref	Expression
1		snssl.1	. . 3 ⊢ 𝐴 ∈ V
2	1	snid 4597	. 2 ⊢ 𝐴 ∈ {𝐴}
3		ssel2 3916	. 2 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
4	2, 3	mpan2 688	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-sn 4562

This theorem is referenced by: (None)