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Theorem snsspw 3509
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 2974 . . 3 (x = AxA)
2 elsn 3365 . . 3 (x {A} ↔ x = A)
3 df-pw 3336 . . . 4 𝒫 A = {xxA}
43abeq2i 2130 . . 3 (x 𝒫 AxA)
51, 2, 43imtr4i 190 . 2 (x {A} → x 𝒫 A)
65ssriv 2926 1 {A} ⊆ 𝒫 A
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  wss 2894  𝒫 cpw 3334  {csn 3350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356
This theorem is referenced by:  snexgOLD  3909  snexg  3910
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