Theorem snsslVD 44827

Description: Virtual deduction proof of snssl 44828. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
snsslVD.1	⊢ 𝐴 ∈ V

Assertion

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

Proof of Theorem snsslVD

Step	Hyp	Ref	Expression
1		idn1 44572	. . 3 ⊢ (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2		snsslVD.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4667	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		ssel2 3990	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
5	1, 3, 4	e10an 44693	. 2 ⊢ (   {𝐴} ⊆ 𝐵   ▶   𝐴 ∈ 𝐵   )
6	5	in1 44569	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-sn 4632  df-vd1 44568

This theorem is referenced by: (None)