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Theorem pw10 4161
 Description: Compute the unit power class of ∅ (Contributed by SF, 22-Jan-2015.)
Assertion
Ref Expression
pw10 1 =

Proof of Theorem pw10
StepHypRef Expression
1 df-pw1 4137 . 2 1 = ( ∩ 1c)
2 pw0 4160 . . 3 = {}
32ineq1i 3453 . 2 ( ∩ 1c) = ({} ∩ 1c)
4 disj 3591 . . 3 (({} ∩ 1c) = x {} ¬ x 1c)
5 0nel1c 4159 . . . 4 ¬ 1c
6 elsn 3748 . . . . 5 (x {} ↔ x = )
7 eleq1 2413 . . . . 5 (x = → (x 1c 1c))
86, 7sylbi 187 . . . 4 (x {} → (x 1c 1c))
95, 8mtbiri 294 . . 3 (x {} → ¬ x 1c)
104, 9mprgbir 2684 . 2 ({} ∩ 1c) =
111, 3, 103eqtri 2377 1 1 =
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  ∅c0 3550  ℘cpw 3722  {csn 3737  1cc1c 4134  ℘1cpw1 4135 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  ax-nin 4078  ax-sn 4087 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-1c 4136  df-pw1 4137 This theorem is referenced by:  pw10b  4166  ncfinraise  4481  tfindi  4496  tfin0c  4497  sfin01  4528  tc0c  6163  tcdi  6164  ce0nnul  6177  ce0addcnnul  6179  ce0nn  6180  ce0nulnc  6184  ce0  6190
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