Theorem pwtp 4834

Description: The power set of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Assertion

Ref	Expression
pwtp	⊢ 𝒫 {𝐴, 𝐵, 𝐶} = (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))

Proof of Theorem pwtp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velpw 4538	. . 3 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶} ↔ 𝑥 ⊆ {𝐴, 𝐵, 𝐶})
2		elun 4083	. . . . . 6 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ (𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}))
3		vex 3436	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	elpr 4584	. . . . . . 7 ⊢ (𝑥 ∈ {∅, {𝐴}} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
5	3	elpr 4584	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐵}, {𝐴, 𝐵}} ↔ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))
6	4, 5	orbi12i 912	. . . . . 6 ⊢ ((𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
7	2, 6	bitri 274	. . . . 5 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
8		elun 4083	. . . . . 6 ⊢ (𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ∨ 𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))
9	3	elpr 4584	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ↔ (𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}))
10	3	elpr 4584	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}} ↔ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))
11	9, 10	orbi12i 912	. . . . . 6 ⊢ ((𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ∨ 𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶})))
12	8, 11	bitri 274	. . . . 5 ⊢ (𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶})))
13	7, 12	orbi12i 912	. . . 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}))))
14		elun 4083	. . . 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}})))
15		sstp 4767	. . . 4 ⊢ (𝑥 ⊆ {𝐴, 𝐵, 𝐶} ↔ (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∨ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))))
16	13, 14, 15	3bitr4ri 304	. . 3 ⊢ (𝑥 ⊆ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})))
17	1, 16	bitri 274	. 2 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})))
18	17	eqriv 2735	1 ⊢ 𝒫 {𝐴, 𝐵, 𝐶} = (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))

Syntax hints:   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  {cpr 4563  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566

This theorem is referenced by:  ex-pw  28793