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Theorem pwsnss 3602
 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 3552 . . 3 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
21ss2abi 3040 . 2 {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥𝑥 ⊆ {𝐴}}
3 dfpr2 3422 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
4 df-pw 3389 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
52, 3, 43sstr4i 3012 1 {∅, {𝐴}} ⊆ 𝒫 {𝐴}
 Colors of variables: wff set class Syntax hints:   ∨ wo 639   = wceq 1259  {cab 2042   ⊆ wss 2945  ∅c0 3252  𝒫 cpw 3387  {csn 3403  {cpr 3404 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410 This theorem is referenced by:  pwpw0ss  3603
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