Theorem pw10 4161

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 4137	. 2 |- ~P1 (/) = (~P(/) i^i 1c)
2		pw0 4160	. . 3 |- ~P(/) = {(/)}
3	2	ineq1i 3453	. 2 |- (~P(/) i^i 1c) = ({(/)} i^i 1c)
4		disj 3591	. . 3 |- (({(/)} i^i 1c) = (/) <-> A.x e. {(/)} -. x e. 1c)
5		0nel1c 4159	. . . 4 |- -. (/) e. 1c
6		elsn 3748	. . . . 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 2684	. 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 3208  (/)c0 3550  ~Pcpw 3722  {csn 3737  1_cc1c 4134  ~P₁ cpw1 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