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Theorem pwsnss 3653
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 3603 . . 3 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
21ss2abi 3094 . 2 {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥𝑥 ⊆ {𝐴}}
3 dfpr2 3469 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
4 df-pw 3435 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
52, 3, 43sstr4i 3066 1 {∅, {𝐴}} ⊆ 𝒫 {𝐴}
Colors of variables: wff set class
Syntax hints:  wo 665   = wceq 1290  {cab 2075  wss 3000  c0 3287  𝒫 cpw 3433  {csn 3450  {cpr 3451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457
This theorem is referenced by:  pwpw0ss  3654
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