Theorem snssl 44105

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 4782. The proof of this theorem was automatically generated from snsslVD 44104 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 4657	. 2 ⊢ 𝐴 ∈ {𝐴}
3		ssel2 3970	. 2 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
4	2, 3	mpan2 688	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3941  {csn 4621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-sn 4622

This theorem is referenced by: (None)