Theorem snsslVD 45565

Description: Virtual deduction proof of snssl 45566. (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 45311	. . 3 ⊢ (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2		snsslVD.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	snid 4627	. . 3 ⊢ 𝐴 ∈ {𝐴}
4		ssel2 3931	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ∧ 𝐴 ∈ {𝐴}) → 𝐴 ∈ 𝐵)
5	1, 3, 4	e10an 45432	. 2 ⊢ (   {𝐴} ⊆ 𝐵   ▶   𝐴 ∈ 𝐵   )
6	5	in1 45308	1 ⊢ ({𝐴} ⊆ 𝐵 → 𝐴 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-ss 3921  df-sn 4589  df-vd1 45307

This theorem is used by: (None)