Theorem snsspw 4643

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 3909	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
2		velsn 4451	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		selpw 4423	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 2, 3	3imtr4i 284	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
5	4	ssriv 3858	1 ⊢ {𝐴} ⊆ 𝒫 𝐴

Syntax hints:   = wceq 1507   ∈ wcel 2048   ⊆ wss 3825  𝒫 cpw 4416  {csn 4435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-in 3832  df-ss 3839  df-pw 4418  df-sn 4436

This theorem is referenced by:  snexALT  5130  snwf  9024  tsksn  9972  mnusnd  39924