Theorem pw1ne1 7541

Description: The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
pw1ne1	|- ~P 1o =/= 1o

Proof of Theorem pw1ne1

Step	Hyp	Ref	Expression
1		pw1on 7538	. . . 4 |- ~P 1o e. On
2	1	onirri 4667	. . 3 |- -. ~P 1o e. ~P 1o
3		df1o2 6663	. . . . 5 |- 1o = { (/) }
4		pwpw0ss 3911	. . . . . . . 8 |- { (/) , { (/) } } C_ ~P { (/) }
5	3	pweqi 3675	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	4, 5	sseqtrri 3275	. . . . . . 7 |- { (/) , { (/) } } C_ ~P 1o
7		0ex 4239	. . . . . . . 8 |- (/) e. _V
8		p0ex 4303	. . . . . . . 8 |- { (/) } e. _V
9	7, 8	prss 3852	. . . . . . 7 |- ( ( (/) e. ~P 1o /\ { (/) } e. ~P 1o ) <-> { (/) , { (/) } } C_ ~P 1o )
10	6, 9	mpbir 146	. . . . . 6 |- ( (/) e. ~P 1o /\ { (/) } e. ~P 1o )
11	10	simpri 113	. . . . 5 |- { (/) } e. ~P 1o
12	3, 11	eqeltri 2307	. . . 4 |- 1o e. ~P 1o
13		eleq1 2297	. . . 4 |- ( ~P 1o = 1o -> ( ~P 1o e. ~P 1o <-> 1o e. ~P 1o ) )
14	12, 13	mpbiri 168	. . 3 |- ( ~P 1o = 1o -> ~P 1o e. ~P 1o )
15	2, 14	mto 668	. 2 |- -. ~P 1o = 1o
16	15	neir 2417	1 |- ~P 1o =/= 1o

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   C_ wss 3213   (/)c0 3510   ~Pcpw 3671   {csn 3691   {cpr 3692   1oc1o 6642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-uni 3917  df-tr 4211  df-iord 4489  df-on 4491  df-suc 4494  df-1o 6649

This theorem is referenced by:  pw1nel3  7543  sucpw1nel3  7545