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Theorem pwtpss 3568
Description: The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)
Assertion
Ref Expression
pwtpss (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) ⊆ 𝒫 {A, B, 𝐶}

Proof of Theorem pwtpss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sstpr 3519 . . 3 ((((x = ∅ x = {A}) (x = {B} x = {A, B})) ((x = {𝐶} x = {A, 𝐶}) (x = {B, 𝐶} x = {A, B, 𝐶}))) → x ⊆ {A, B, 𝐶})
2 elun 3078 . . . 4 (x (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) ↔ (x ({∅, {A}} ∪ {{B}, {A, B}}) x ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})))
3 elun 3078 . . . . . 6 (x ({∅, {A}} ∪ {{B}, {A, B}}) ↔ (x {∅, {A}} x {{B}, {A, B}}))
4 vex 2554 . . . . . . . 8 x V
54elpr 3385 . . . . . . 7 (x {∅, {A}} ↔ (x = ∅ x = {A}))
64elpr 3385 . . . . . . 7 (x {{B}, {A, B}} ↔ (x = {B} x = {A, B}))
75, 6orbi12i 680 . . . . . 6 ((x {∅, {A}} x {{B}, {A, B}}) ↔ ((x = ∅ x = {A}) (x = {B} x = {A, B})))
83, 7bitri 173 . . . . 5 (x ({∅, {A}} ∪ {{B}, {A, B}}) ↔ ((x = ∅ x = {A}) (x = {B} x = {A, B})))
9 elun 3078 . . . . . 6 (x ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}}) ↔ (x {{𝐶}, {A, 𝐶}} x {{B, 𝐶}, {A, B, 𝐶}}))
104elpr 3385 . . . . . . 7 (x {{𝐶}, {A, 𝐶}} ↔ (x = {𝐶} x = {A, 𝐶}))
114elpr 3385 . . . . . . 7 (x {{B, 𝐶}, {A, B, 𝐶}} ↔ (x = {B, 𝐶} x = {A, B, 𝐶}))
1210, 11orbi12i 680 . . . . . 6 ((x {{𝐶}, {A, 𝐶}} x {{B, 𝐶}, {A, B, 𝐶}}) ↔ ((x = {𝐶} x = {A, 𝐶}) (x = {B, 𝐶} x = {A, B, 𝐶})))
139, 12bitri 173 . . . . 5 (x ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}}) ↔ ((x = {𝐶} x = {A, 𝐶}) (x = {B, 𝐶} x = {A, B, 𝐶})))
148, 13orbi12i 680 . . . 4 ((x ({∅, {A}} ∪ {{B}, {A, B}}) x ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) ↔ (((x = ∅ x = {A}) (x = {B} x = {A, B})) ((x = {𝐶} x = {A, 𝐶}) (x = {B, 𝐶} x = {A, B, 𝐶}))))
152, 14bitri 173 . . 3 (x (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) ↔ (((x = ∅ x = {A}) (x = {B} x = {A, B})) ((x = {𝐶} x = {A, 𝐶}) (x = {B, 𝐶} x = {A, B, 𝐶}))))
164elpw 3357 . . 3 (x 𝒫 {A, B, 𝐶} ↔ x ⊆ {A, B, 𝐶})
171, 15, 163imtr4i 190 . 2 (x (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) → x 𝒫 {A, B, 𝐶})
1817ssriv 2943 1 (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) ⊆ 𝒫 {A, B, 𝐶}
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  {ctp 3369
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-3or 885  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  df-tp 3375
This theorem is referenced by: (None)
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