Theorem snsslVD 45520

Description: Virtual deduction proof of snssl 45521. (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 45266	. . 3 ⊢ (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2		snsslVD.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4629	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		ssel2 3933	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
5	1, 3, 4	e10an 45387	. 2 ⊢ (   {𝐴} ⊆ 𝐵   ▶   𝐴 ∈ 𝐵   )
6	5	in1 45263	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3906  {csn 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3923  df-sn 4591  df-vd1 45262

This theorem is referenced by: (None)