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Theorem snsspw 3526
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 2991 . . 3 (x = AxA)
2 elsn 3382 . . 3 (x {A} ↔ x = A)
3 df-pw 3353 . . . 4 𝒫 A = {xxA}
43abeq2i 2145 . . 3 (x 𝒫 AxA)
51, 2, 43imtr4i 190 . 2 (x {A} → x 𝒫 A)
65ssriv 2943 1 {A} ⊆ 𝒫 A
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  wss 2911  𝒫 cpw 3351  {csn 3367
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373
This theorem is referenced by:  snexgOLD  3926  snexg  3927
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