Theorem snssiALT 44848

Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4808. This theorem was automatically generated from snssiALTVD 44847 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem snssiALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 4642	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2		eleq1a 2836	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
3	1, 2	biimtrid 242	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
4	3	alrimiv 1927	. 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
5		df-ss 3968	. 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
6	4, 5	sylibr 234	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  {csn 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-sn 4627

This theorem is referenced by: (None)