Theorem pw1ne1 7578

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 7575	. . . 4 |- ~P 1o e. On
2	1	onirri 4685	. . 3 |- -. ~P 1o e. ~P 1o
3		df1o2 6691	. . . . 5 |- 1o = { (/) }
4		pwpw0ss 3925	. . . . . . . 8 |- { (/) , { (/) } } C_ ~P { (/) }
5	3	pweqi 3689	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	4, 5	sseqtrri 3283	. . . . . . 7 |- { (/) , { (/) } } C_ ~P 1o
7		0ex 4255	. . . . . . . 8 |- (/) e. _V
8		p0ex 4320	. . . . . . . 8 |- { (/) } e. _V
9	7, 8	prss 3866	. . . . . . 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 2311	. . . 4 |- 1o e. ~P 1o
13		eleq1 2301	. . . 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 672	. 2 |- -. ~P 1o = 1o
16	15	neir 2423	1 |- ~P 1o =/= 1o

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   C_ wss 3220   (/)c0 3520   ~Pcpw 3685   {csn 3705   {cpr 3706   1oc1o 6670

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-1o 6677

This theorem is referenced by:  pw1nel3  7580  sucpw1nel3  7582