Theorem pw1ne1 7446

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 7443	. . . 4 |- ~P 1o e. On
2	1	onirri 4641	. . 3 |- -. ~P 1o e. ~P 1o
3		df1o2 6595	. . . . 5 |- 1o = { (/) }
4		pwpw0ss 3888	. . . . . . . 8 |- { (/) , { (/) } } C_ ~P { (/) }
5	3	pweqi 3656	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	4, 5	sseqtrri 3262	. . . . . . 7 |- { (/) , { (/) } } C_ ~P 1o
7		0ex 4216	. . . . . . . 8 |- (/) e. _V
8		p0ex 4278	. . . . . . . 8 |- { (/) } e. _V
9	7, 8	prss 3829	. . . . . . 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 2304	. . . 4 |- 1o e. ~P 1o
13		eleq1 2294	. . . 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 2405	1 |- ~P 1o =/= 1o

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   =/= wne 2402   C_ wss 3200   (/)c0 3494   ~Pcpw 3652   {csn 3669   {cpr 3670   1oc1o 6574

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468  df-1o 6581

This theorem is referenced by:  pw1nel3  7448  sucpw1nel3  7450