Theorem snsspw 4813

Description: The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.)

Assertion

Ref	Expression
snsspw	⊢ {𝐴} ⊆ 𝒫 𝐴

Proof of Theorem snsspw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqimss 4003	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
2		velsn 4610	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velpw 4572	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 2, 3	3imtr4i 295	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
5	4	ssriv 3949	1 ⊢ {𝐴} ⊆ 𝒫 𝐴

Syntax hints:   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  𝒫 cpw 4567  {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-pw 4569  df-sn 4595

This theorem is referenced by:  snexALT  5355  snwf  9780  tsksn  10744  mnusnd  44869