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Theorem pwsnss 3624
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 3574 . . 3 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
21ss2abi 3079 . 2 {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥𝑥 ⊆ {𝐴}}
3 dfpr2 3444 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
4 df-pw 3411 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
52, 3, 43sstr4i 3051 1 {∅, {𝐴}} ⊆ 𝒫 {𝐴}
Colors of variables: wff set class
Syntax hints:  wo 662   = wceq 1287  {cab 2071  wss 2986  c0 3272  𝒫 cpw 3409  {csn 3425  {cpr 3426
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-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432
This theorem is referenced by:  pwpw0ss  3625
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