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Theorem snsspw 3779
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 3224 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 3624 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 df-pw 3592 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
43abeq2i 2300 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
51, 2, 43imtr4i 201 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
65ssriv 3174 1 {𝐴} ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  wss 3144  𝒫 cpw 3590  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613
This theorem is referenced by:  snexg  4202
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