Theorem pw1ne1 7414

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 7411	. . . 4 |- ~P 1o e. On
2	1	onirri 4635	. . 3 |- -. ~P 1o e. ~P 1o
3		df1o2 6575	. . . . 5 |- 1o = { (/) }
4		pwpw0ss 3883	. . . . . . . 8 |- { (/) , { (/) } } C_ ~P { (/) }
5	3	pweqi 3653	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	4, 5	sseqtrri 3259	. . . . . . 7 |- { (/) , { (/) } } C_ ~P 1o
7		0ex 4211	. . . . . . . 8 |- (/) e. _V
8		p0ex 4272	. . . . . . . 8 |- { (/) } e. _V
9	7, 8	prss 3824	. . . . . . 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 2302	. . . 4 |- 1o e. ~P 1o
13		eleq1 2292	. . . 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 666	. 2 |- -. ~P 1o = 1o
16	15	neir 2403	1 |- ~P 1o =/= 1o

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   C_ wss 3197   (/)c0 3491   ~Pcpw 3649   {csn 3666   {cpr 3667   1oc1o 6555

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462  df-1o 6562

This theorem is referenced by:  pw1nel3  7416  sucpw1nel3  7418