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Theorem pwpw0 3855
 Description: Compute the power set of the power set of the empty set. (See pw0 4160 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3881, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
pwpw0 {} = {, {}}

Proof of Theorem pwpw0
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3262 . . . . . . . . 9 (x {} ↔ y(y xy {}))
2 elsn 3748 . . . . . . . . . . 11 (y {} ↔ y = )
32imbi2i 303 . . . . . . . . . 10 ((y xy {}) ↔ (y xy = ))
43albii 1566 . . . . . . . . 9 (y(y xy {}) ↔ y(y xy = ))
51, 4bitri 240 . . . . . . . 8 (x {} ↔ y(y xy = ))
6 neq0 3560 . . . . . . . . . 10 x = y y x)
7 exintr 1614 . . . . . . . . . 10 (y(y xy = ) → (y y xy(y x y = )))
86, 7syl5bi 208 . . . . . . . . 9 (y(y xy = ) → (¬ x = y(y x y = )))
9 exancom 1586 . . . . . . . . . . 11 (y(y x y = ) ↔ y(y = y x))
10 df-clel 2349 . . . . . . . . . . 11 ( xy(y = y x))
119, 10bitr4i 243 . . . . . . . . . 10 (y(y x y = ) ↔ x)
12 snssi 3852 . . . . . . . . . 10 ( x → {} x)
1311, 12sylbi 187 . . . . . . . . 9 (y(y x y = ) → {} x)
148, 13syl6 29 . . . . . . . 8 (y(y xy = ) → (¬ x = → {} x))
155, 14sylbi 187 . . . . . . 7 (x {} → (¬ x = → {} x))
1615anc2li 540 . . . . . 6 (x {} → (¬ x = → (x {} {} x)))
17 eqss 3287 . . . . . 6 (x = {} ↔ (x {} {} x))
1816, 17syl6ibr 218 . . . . 5 (x {} → (¬ x = x = {}))
1918orrd 367 . . . 4 (x {} → (x = x = {}))
20 0ss 3579 . . . . . 6 {}
21 sseq1 3292 . . . . . 6 (x = → (x {} ↔ {}))
2220, 21mpbiri 224 . . . . 5 (x = x {})
23 eqimss 3323 . . . . 5 (x = {} → x {})
2422, 23jaoi 368 . . . 4 ((x = x = {}) → x {})
2519, 24impbii 180 . . 3 (x {} ↔ (x = x = {}))
2625abbii 2465 . 2 {x x {}} = {x (x = x = {})}
27 df-pw 3724 . 2 {} = {x x {}}
28 dfpr2 3749 . 2 {, {}} = {x (x = x = {})}
2926, 27, 283eqtr4i 2383 1 {} = {, {}}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ 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-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: (None)
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