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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.
StepHypRef Expression
1 eqimss 3909 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 4451 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 selpw 4423 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 2, 33imtr4i 284 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
54ssriv 3858 1 {𝐴} ⊆ 𝒫 𝐴
 Colors of variables: wff setvar class 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
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