Theorem snssiALTVD 44798

Description: Virtual deduction proof of snssiALT 44799. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem snssiALTVD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3993	. . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
2		idn1 44545	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		idn2 44584	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4		velsn 4664	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5	3, 4	e2bi 44603	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6		eleq1a 2839	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	2, 5, 6	e12 44695	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ 𝐵   )
8	7	in2 44576	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)   )
9	8	gen11 44587	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)   )
10		biimpr 220	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) → {𝐴} ⊆ 𝐵))
11	1, 9, 10	e01 44662	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
12	11	in1 44542	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-sn 4649  df-vd1 44541  df-vd2 44549

This theorem is referenced by: (None)