Theorem snsslVD 43575

Description: Virtual deduction proof of snssl 43576. (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 43320	. . 3 ⊢ (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2		snsslVD.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4663	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		ssel2 3976	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
5	1, 3, 4	e10an 43441	. 2 ⊢ (   {𝐴} ⊆ 𝐵   ▶   𝐴 ∈ 𝐵   )
6	5	in1 43317	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3947  {csn 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-sn 4628  df-vd1 43316

This theorem is referenced by: (None)