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Theorem snsspw 4366
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 3649 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 4184 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 selpw 4156 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
41, 2, 33imtr4i 281 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
54ssriv 3599 1 {𝐴} ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wcel 1988  wss 3567  𝒫 cpw 4149  {csn 4168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-in 3574  df-ss 3581  df-pw 4151  df-sn 4169
This theorem is referenced by:  snexALT  4843  snwf  8657  tsksn  9567
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