Theorem snssd 3854

Description: The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
snssd.1	⊢ (φ → A ∈ B)

Assertion

Ref	Expression
snssd	⊢ (φ → {A} ⊆ B)

Proof of Theorem snssd

Step	Hyp	Ref	Expression
1		snssd.1	. 2 ⊢ (φ → A ∈ B)
2		snssg 3845	. . 3 ⊢ (A ∈ B → (A ∈ B ↔ {A} ⊆ B))
3	1, 2	syl 15	. 2 ⊢ (φ → (A ∈ B ↔ {A} ⊆ B))
4	1, 3	mpbid 201	1 ⊢ (φ → {A} ⊆ B)

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710   ⊆ wss 3258  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-sn 3742

This theorem is referenced by:  nchoicelem6  6295  frecsuc  6323