Theorem pwtp 4431

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		selpw 4165	. . 3 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶} ↔ 𝑥 ⊆ {𝐴, 𝐵, 𝐶})
2		elun 3753	. . . . . 6 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ (𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}))
3		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	3	elpr 4198	. . . . . . 7 ⊢ (𝑥 ∈ {∅, {𝐴}} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
5	3	elpr 4198	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐵}, {𝐴, 𝐵}} ↔ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))
6	4, 5	orbi12i 543	. . . . . 6 ⊢ ((𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
7	2, 6	bitri 264	. . . . 5 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
8		elun 3753	. . . . . 6 ⊢ (𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ∨ 𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))
9	3	elpr 4198	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ↔ (𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}))
10	3	elpr 4198	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}} ↔ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))
11	9, 10	orbi12i 543	. . . . . 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 264	. . . . 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 543	. . . 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 3753	. . . 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 4367	. . . 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 293	. . 3 ⊢ (𝑥 ⊆ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})))
17	1, 16	bitri 264	. 2 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})))
18	17	eqriv 2619	1 ⊢ 𝒫 {𝐴, 𝐵, 𝐶} = (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))

Syntax hints:   ∨ wo 383   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  {cpr 4179  {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  ex-pw  27286