Theorem snssiALT 45462

Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4756. This theorem was automatically generated from snssiALTVD 45461 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 4610	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2		eleq1a 2864	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
3	1, 2	biimtrid 245	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
4	3	alrimiv 1954	. 2 ⊢ (𝐴 ∈ 𝐵 → ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
5		df-ss 3930	. 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
6	4, 5	sylibr 237	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1565   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-sn 4595

This theorem is referenced by: (None)