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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.
StepHypRef Expression
1 eqimss 4035 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 4646 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velpw 4609 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 2, 33imtr4i 291 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
54ssriv 3980 1 {𝐴} ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
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
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