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Theorem pwpr 3883
 Description: The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
Assertion
Ref Expression
pwpr {A, B} = ({, {A}} ∪ {{B}, {A, B}})

Proof of Theorem pwpr
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sspr 3869 . . . 4 (x {A, B} ↔ ((x = x = {A}) (x = {B} x = {A, B})))
2 vex 2862 . . . . . 6 x V
32elpr 3751 . . . . 5 (x {, {A}} ↔ (x = x = {A}))
42elpr 3751 . . . . 5 (x {{B}, {A, B}} ↔ (x = {B} x = {A, B}))
53, 4orbi12i 507 . . . 4 ((x {, {A}} x {{B}, {A, B}}) ↔ ((x = x = {A}) (x = {B} x = {A, B})))
61, 5bitr4i 243 . . 3 (x {A, B} ↔ (x {, {A}} x {{B}, {A, B}}))
72elpw 3728 . . 3 (x {A, B} ↔ x {A, B})
8 elun 3220 . . 3 (x ({, {A}} ∪ {{B}, {A, B}}) ↔ (x {, {A}} x {{B}, {A, B}}))
96, 7, 83bitr4i 268 . 2 (x {A, B} ↔ x ({, {A}} ∪ {{B}, {A, B}}))
109eqriv 2350 1 {A, B} = ({, {A}} ∪ {{B}, {A, B}})
 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 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 This theorem is referenced by:  pwpwpw0  3885
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