Theorem snssiALT 43574

Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4810. This theorem was automatically generated from snssiALTVD 43573 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 4643	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2		eleq1a 2828	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
3	1, 2	biimtrid 241	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
4	3	alrimiv 1930	. 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
5		dfss2 3967	. 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
6	4, 5	sylibr 233	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541   ∈ wcel 2106   ⊆ wss 3947  {csn 4627

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 3954  df-ss 3964  df-sn 4628

This theorem is referenced by: (None)