Theorem pwtp 3884

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

Assertion

Ref	Expression
pwtp	⊢ ℘{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		vex 2862	. . . 4 ⊢ x ∈ V
2	1	elpw 3728	. . 3 ⊢ (x ∈ ℘{A, B, C} ↔ x ⊆ {A, B, C})
3		elun 3220	. . . . . 6 ⊢ (x ∈ ({∅, {A}} ∪ {{B}, {A, B}}) ↔ (x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}}))
4	1	elpr 3751	. . . . . . 7 ⊢ (x ∈ {∅, {A}} ↔ (x = ∅ ∨ x = {A}))
5	1	elpr 3751	. . . . . . 7 ⊢ (x ∈ {{B}, {A, B}} ↔ (x = {B} ∨ x = {A, B}))
6	4, 5	orbi12i 507	. . . . . 6 ⊢ ((x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}}) ↔ ((x = ∅ ∨ x = {A}) ∨ (x = {B} ∨ x = {A, B})))
7	3, 6	bitri 240	. . . . 5 ⊢ (x ∈ ({∅, {A}} ∪ {{B}, {A, B}}) ↔ ((x = ∅ ∨ x = {A}) ∨ (x = {B} ∨ x = {A, B})))
8		elun 3220	. . . . . 6 ⊢ (x ∈ ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}}) ↔ (x ∈ {{C}, {A, C}} ∨ x ∈ {{B, C}, {A, B, C}}))
9	1	elpr 3751	. . . . . . 7 ⊢ (x ∈ {{C}, {A, C}} ↔ (x = {C} ∨ x = {A, C}))
10	1	elpr 3751	. . . . . . 7 ⊢ (x ∈ {{B, C}, {A, B, C}} ↔ (x = {B, C} ∨ x = {A, B, C}))
11	9, 10	orbi12i 507	. . . . . 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 240	. . . . 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 507	. . . 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 3220	. . . 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 3870	. . . 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 269	. . 3 ⊢ (x ⊆ {A, B, C} ↔ x ∈ (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}})))
17	2, 16	bitri 240	. 2 ⊢ (x ∈ ℘{A, B, C} ↔ x ∈ (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}})))
18	17	eqriv 2350	1 ⊢ ℘{A, B, C} = (({∅, {A}} ∪ {{B}, {A, B}}) ∪ ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}}))

Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ cun 3207   ⊆ wss 3257  ∅c0 3550  ℘cpw 3722  {csn 3737  {cpr 3738  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by: (None)