Theorem snsspw 4849

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 4054	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
2		velsn 4647	. . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velpw 4610	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 2, 3	3imtr4i 292	. 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
5	4	ssriv 3999	1 ⊢ {𝐴} ⊆ 𝒫 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-pw 4607  df-sn 4632

This theorem is referenced by:  snexALT  5389  snwf  9847  tsksn  10798  mnusnd  44264