Theorem sucpw1nel3 7293

Description: The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
sucpw1nel3	|- -. suc ~P 1o e. 3o

Proof of Theorem sucpw1nel3

Step	Hyp	Ref	Expression
1		1oex 6477	. . . . . . 7 |- 1o e. _V
2	1	pwex 4212	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4448	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3456	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7288	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3634	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2453	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6482	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2260	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 672	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2257	. . . . . . 7 |- ( suc ~P 1o = 1o -> ( ~P 1o e. suc ~P 1o <-> ~P 1o e. 1o ) )
12	3, 11	mpbii 148	. . . . . 6 |- ( suc ~P 1o = 1o -> ~P 1o e. 1o )
13	10, 12	mto 663	. . . . 5 |- -. suc ~P 1o = 1o
14	13	neir 2367	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3642	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6483	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2260	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 672	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7289	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3642	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2260	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 672	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2257	. . . . 5 |- ( suc ~P 1o = 2o -> ( ~P 1o e. suc ~P 1o <-> ~P 1o e. 2o ) )
24	3, 23	mpbii 148	. . . 4 |- ( suc ~P 1o = 2o -> ~P 1o e. 2o )
25	22, 24	mto 663	. . 3 |- -. suc ~P 1o = 2o
26	2	sucex 4531	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3634	. . 3 |- ( suc ~P 1o e. { 2o } <-> suc ~P 1o = 2o )
28	25, 27	mtbir 672	. 2 |- -. suc ~P 1o e. { 2o }
29		ioran 753	. . 3 |- ( -. ( suc ~P 1o e. 2o \/ suc ~P 1o e. { 2o } ) <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
30		df-3o 6471	. . . . . 6 |- 3o = suc 2o
31		df-suc 4402	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2214	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2260	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3300	. . . 4 |- ( suc ~P 1o e. ( 2o u. { 2o } ) <-> ( suc ~P 1o e. 2o \/ suc ~P 1o e. { 2o } ) )
35	33, 34	bitri 184	. . 3 |- ( suc ~P 1o e. 3o <-> ( suc ~P 1o e. 2o \/ suc ~P 1o e. { 2o } ) )
36	29, 35	xchnxbir 682	. 2 |- ( -. suc ~P 1o e. 3o <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
37	18, 28, 36	mpbir2an 944	1 |- -. suc ~P 1o e. 3o

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   u. cun 3151   (/)c0 3446   ~Pcpw 3601   {csn 3618   {cpr 3619   suc csuc 4396   1oc1o 6462   2oc2o 6463   3oc3o 6464

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-1o 6469  df-2o 6470  df-3o 6471

This theorem is referenced by:  onntri35  7297