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Theorem pwsnss 3574
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 3524 . . 3 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
21ss2abi 3012 . 2 {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥𝑥 ⊆ {𝐴}}
3 dfpr2 3394 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
4 df-pw 3361 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
52, 3, 43sstr4i 2984 1 {∅, {𝐴}} ⊆ 𝒫 {𝐴}
Colors of variables: wff set class
Syntax hints:  wo 629   = wceq 1243  {cab 2026  wss 2917  c0 3224  𝒫 cpw 3359  {csn 3375  {cpr 3376
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382
This theorem is referenced by:  pwpw0ss  3575
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