Theorem pw10 4162

Description: Compute the unit power class of (/). (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
pw10	|- ~P1 (/) = (/)

Proof of Theorem pw10

Step	Hyp	Ref	Expression
1		df-pw1 4138	. 2 |- ~P1 (/) = (~P(/) i^i 1c)
2		pw0 4161	. . 3 |- ~P(/) = {(/)}
3	2	ineq1i 3454	. 2 |- (~P(/) i^i 1c) = ({(/)} i^i 1c)
4		disj 3592	. . 3 |- (({(/)} i^i 1c) = (/) <-> A.x e. {(/)} -. x e. 1c)
5		0nel1c 4160	. . . 4 |- -. (/) e. 1c
6		elsn 3749	. . . . 5 |- (x e. {(/)} <-> x = (/))
7		eleq1 2413	. . . . 5 |- (x = (/) -> (x e. 1c <-> (/) e. 1c))
8	6, 7	sylbi 187	. . . 4 |- (x e. {(/)} -> (x e. 1c <-> (/) e. 1c))
9	5, 8	mtbiri 294	. . 3 |- (x e. {(/)} -> -. x e. 1c)
10	4, 9	mprgbir 2685	. 2 |- ({(/)} i^i 1c) = (/)
11	1, 3, 10	3eqtri 2377	1 |- ~P1 (/) = (/)

Syntax hints:  -. wn 3   <-> wb 176   = wceq 1642   e. wcel 1710   i^i cin 3209  (/)c0 3551  ~Pcpw 3723  {csn 3738  1_cc1c 4135  ~P₁ cpw1 4136

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-1c 4137  df-pw1 4138

This theorem is referenced by:  pw10b  4167  ncfinraise  4482  tfindi  4497  tfin0c  4498  sfin01  4529  tc0c  6164  tcdi  6165  ce0nnul  6178  ce0addcnnul  6180  ce0nn  6181  ce0nulnc  6185  ce0  6191