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

 Description: The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
Assertion
Ref Expression
pwprss ({∅, {A}} ∪ {{B}, {A, B}}) ⊆ 𝒫 {A, B}

Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . 6 x V
21elpr 3385 . . . . 5 (x {∅, {A}} ↔ (x = ∅ x = {A}))
31elpr 3385 . . . . 5 (x {{B}, {A, B}} ↔ (x = {B} x = {A, B}))
42, 3orbi12i 680 . . . 4 ((x {∅, {A}} x {{B}, {A, B}}) ↔ ((x = ∅ x = {A}) (x = {B} x = {A, B})))
5 ssprr 3518 . . . 4 (((x = ∅ x = {A}) (x = {B} x = {A, B})) → x ⊆ {A, B})
64, 5sylbi 114 . . 3 ((x {∅, {A}} x {{B}, {A, B}}) → x ⊆ {A, B})
7 elun 3078 . . 3 (x ({∅, {A}} ∪ {{B}, {A, B}}) ↔ (x {∅, {A}} x {{B}, {A, B}}))
81elpw 3357 . . 3 (x 𝒫 {A, B} ↔ x ⊆ {A, B})
96, 7, 83imtr4i 190 . 2 (x ({∅, {A}} ∪ {{B}, {A, B}}) → x 𝒫 {A, B})
109ssriv 2943 1 ({∅, {A}} ∪ {{B}, {A, B}}) ⊆ 𝒫 {A, B}
 Colors of variables: wff set class Syntax hints:   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ∪ cun 2909   ⊆ wss 2911  ∅c0 3218  𝒫 cpw 3351  {csn 3367  {cpr 3368 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374 This theorem is referenced by:  pwpwpw0ss  3569  ord3ex  3932
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