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Theorem snsspw 3603
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 3076 . . 3 (𝑥 = 𝐴𝑥𝐴)
2 velsn 3458 . . 3 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 df-pw 3427 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
43abeq2i 2198 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
51, 2, 43imtr4i 199 . 2 (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴)
65ssriv 3027 1 {𝐴} ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  wss 2997  𝒫 cpw 3425  {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447
This theorem is referenced by:  snexg  4010
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