Theorem snssl 42064

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 4685. The proof of this theorem was automatically generated from snsslVD 42063 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 4563	. 2 ⊢ 𝐴 ∈ {𝐴}
3		ssel2 3882	. 2 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
4	2, 3	mpan2 691	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2112  Vcvv 3398   ⊆ wss 3853  {csn 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-in 3860  df-ss 3870  df-sn 4528

This theorem is referenced by: (None)