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.

Step	Hyp	Ref	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
5	4	elpr 3385	. . . . . . 7 ⊢ (x ∈ {∅, {A}} ↔ (x = ∅ ∨ x = {A}))
6	4	elpr 3385	. . . . . . 7 ⊢ (x ∈ {{B}, {A, B}} ↔ (x = {B} ∨ x = {A, B}))
7	5, 6	orbi12i 680	. . . . . 6 ⊢ ((x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}}) ↔ ((x = ∅ ∨ x = {A}) ∨ (x = {B} ∨ x = {A, B})))
8	3, 7	bitri 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}}))
10	4	elpr 3385	. . . . . . 7 ⊢ (x ∈ {{𝐶}, {A, 𝐶}} ↔ (x = {𝐶} ∨ x = {A, 𝐶}))
11	4	elpr 3385	. . . . . . 7 ⊢ (x ∈ {{B, 𝐶}, {A, B, 𝐶}} ↔ (x = {B, 𝐶} ∨ x = {A, B, 𝐶}))
12	10, 11	orbi12i 680	. . . . . 6 ⊢ ((x ∈ {{𝐶}, {A, 𝐶}} ∨ x ∈ {{B, 𝐶}, {A, B, 𝐶}}) ↔ ((x = {𝐶} ∨ x = {A, 𝐶}) ∨ (x = {B, 𝐶} ∨ x = {A, B, 𝐶})))
13	9, 12	bitri 173	. . . . 5 ⊢ (x ∈ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}}) ↔ ((x = {𝐶} ∨ x = {A, 𝐶}) ∨ (x = {B, 𝐶} ∨ x = {A, B, 𝐶})))
14	8, 13	orbi12i 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}))))
15	2, 14	bitri 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}))))
16	4	elpw 3357	. . 3 ⊢ (x ∈ 𝒫 {A, B, 𝐶} ↔ x ⊆ {A, B, 𝐶})
17	1, 15, 16	3imtr4i 190	. 2 ⊢ (x ∈ (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) → x ∈ 𝒫 {A, B, 𝐶})
18	17	ssriv 2943	1 ⊢ (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{𝐶}, {A, 𝐶}} ∪ {{B, 𝐶}, {A, B, 𝐶}})) ⊆ 𝒫 {A, B, 𝐶}

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-bndl 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)