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Theorem snsspw 4315
 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 3620 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 4141 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 selpw 4115 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 2, 33imtr4i 280 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
54ssriv 3572 1 {𝐴} ⊆ 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-sn 4126 This theorem is referenced by:  snexALT  4778  snwf  8555  tsksn  9461
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