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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.
StepHypRef Expression
1 vex 2862 . . . 4 x V
21elpw 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}}))
41elpr 3751 . . . . . . 7 (x {, {A}} ↔ (x = x = {A}))
51elpr 3751 . . . . . . 7 (x {{B}, {A, B}} ↔ (x = {B} x = {A, B}))
64, 5orbi12i 507 . . . . . 6 ((x {, {A}} x {{B}, {A, B}}) ↔ ((x = x = {A}) (x = {B} x = {A, B})))
73, 6bitri 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}}))
91elpr 3751 . . . . . . 7 (x {{C}, {A, C}} ↔ (x = {C} x = {A, C}))
101elpr 3751 . . . . . . 7 (x {{B, C}, {A, B, C}} ↔ (x = {B, C} x = {A, B, C}))
119, 10orbi12i 507 . . . . . 6 ((x {{C}, {A, C}} x {{B, C}, {A, B, C}}) ↔ ((x = {C} x = {A, C}) (x = {B, C} x = {A, B, C})))
128, 11bitri 240 . . . . 5 (x ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}}) ↔ ((x = {C} x = {A, C}) (x = {B, C} x = {A, B, C})))
137, 12orbi12i 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}))))
1613, 14, 153bitr4ri 269 . . 3 (x {A, B, C} ↔ x (({, {A}} ∪ {{B}, {A, B}}) ∪ ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}})))
172, 16bitri 240 . 2 (x {A, B, C} ↔ x (({, {A}} ∪ {{B}, {A, B}}) ∪ ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}})))
1817eqriv 2350 1 {A, B, C} = (({, {A}} ∪ {{B}, {A, B}}) ∪ ({{C}, {A, C}} ∪ {{B, C}, {A, B, C}}))
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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)
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