Theorem snssiALTVD 43670

Description: Virtual deduction proof of snssiALT 43671. (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		dfss2 3968	. . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
2		idn1 43417	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		idn2 43456	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4		velsn 4644	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5	3, 4	e2bi 43475	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6		eleq1a 2828	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	2, 5, 6	e12 43567	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ 𝐵   )
8	7	in2 43448	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)   )
9	8	gen11 43459	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)   )
10		biimpr 219	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) → {𝐴} ⊆ 𝐵))
11	1, 9, 10	e01 43534	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
12	11	in1 43414	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541   ∈ wcel 2106   ⊆ wss 3948  {csn 4628

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 3955  df-ss 3965  df-sn 4629  df-vd1 43413  df-vd2 43421

This theorem is referenced by: (None)