Theorem snssl 45579

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 4755. The proof of this theorem was automatically generated from snsslVD 45578 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 4633	. 2 ⊢ 𝐴 ∈ {𝐴}
3		ssel2 3935	. 2 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
4	2, 3	mpan2 704	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3458   ⊆ wss 3908  {csn 4594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-ss 3925  df-sn 4595

This theorem is used by: (None)