Theorem snssiALTVD 38084

Description: Virtual deduction proof of snssiALT 38085. (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 3557	. . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
2		idn1 37811	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		idn2 37859	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4		velsn 4141	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5	3, 4	e2bi 37878	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6		eleq1a 2683	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	2, 5, 6	e12 37972	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ 𝐵   )
8	7	in2 37851	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)   )
9	8	gen11 37862	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)   )
10		biimpr 209	. . 3 ⊢ (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) → {𝐴} ⊆ 𝐵))
11	1, 9, 10	e01 37937	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
12	11	in1 37808	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-sn 4126  df-vd1 37807  df-vd2 37815

This theorem is referenced by: (None)