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Theorem snsspw 3691
 Description: The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-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 3151 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 3544 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 df-pw 3512 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
43abeq2i 2250 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
51, 2, 43imtr4i 200 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
65ssriv 3101 1 {𝐴} ⊆ 𝒫 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∈ wcel 1480   ⊆ wss 3071  𝒫 cpw 3510  {csn 3527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533 This theorem is referenced by:  snexg  4108
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