Theorem snsspw 4847

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 4035	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
2		velsn 4646	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velpw 4609	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 2, 3	3imtr4i 291	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
5	4	ssriv 3980	1 ⊢ {𝐴} ⊆ 𝒫 𝐴

Syntax hints:   = wceq 1533   ∈ wcel 2098   ⊆ wss 3944  𝒫 cpw 4604  {csn 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-ss 3961  df-pw 4606  df-sn 4631

This theorem is referenced by:  snexALT  5383  snwf  9834  tsksn  10785  mnusnd  43847