ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwsnss GIF version

Theorem pwsnss 3698
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 3648 . . 3 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
21ss2abi 3137 . 2 {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥𝑥 ⊆ {𝐴}}
3 dfpr2 3514 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
4 df-pw 3480 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
52, 3, 43sstr4i 3106 1 {∅, {𝐴}} ⊆ 𝒫 {𝐴}
Colors of variables: wff set class
Syntax hints:  wo 680   = wceq 1314  {cab 2101  wss 3039  c0 3331  𝒫 cpw 3478  {csn 3495  {cpr 3496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502
This theorem is referenced by:  pwpw0ss  3699
  Copyright terms: Public domain W3C validator