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Theorem pwsnss 3565
 Description: The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
Assertion
Ref Expression
pwsnss {∅, {A}} ⊆ 𝒫 {A}

Proof of Theorem pwsnss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sssnr 3515 . . 3 ((x = ∅ x = {A}) → x ⊆ {A})
21ss2abi 3006 . 2 {x ∣ (x = ∅ x = {A})} ⊆ {xx ⊆ {A}}
3 dfpr2 3383 . 2 {∅, {A}} = {x ∣ (x = ∅ x = {A})}
4 df-pw 3353 . 2 𝒫 {A} = {xx ⊆ {A}}
52, 3, 43sstr4i 2978 1 {∅, {A}} ⊆ 𝒫 {A}
 Colors of variables: wff set class Syntax hints:   ∨ wo 628   = wceq 1242  {cab 2023   ⊆ wss 2911  ∅c0 3218  𝒫 cpw 3351  {csn 3367  {cpr 3368 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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374 This theorem is referenced by:  pwpw0ss  3566
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