Theorem snsslVD 44147

Description: Virtual deduction proof of snssl 44148. (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 43892	. . 3 ⊢ (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2		snsslVD.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4659	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		ssel2 3972	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
5	1, 3, 4	e10an 44013	. 2 ⊢ (   {𝐴} ⊆ 𝐵   ▶   𝐴 ∈ 𝐵   )
6	5	in1 43889	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3468   ⊆ wss 3943  {csn 4623

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-sn 4624  df-vd1 43888

This theorem is referenced by: (None)