Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsspw Structured version   Visualization version   GIF version

Theorem snsspw 4738
 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.
StepHypRef Expression
1 eqimss 3974 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 4544 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velpw 4505 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 2, 33imtr4i 295 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
54ssriv 3922 1 {𝐴} ⊆ 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2112   ⊆ wss 3884  𝒫 cpw 4500  {csn 4528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529 This theorem is referenced by:  snexALT  5252  snwf  9226  tsksn  10175  mnusnd  40963
 Copyright terms: Public domain W3C validator