Theorem sucpw1nel3 7316

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 6491	. . . . . . 7 |- 1o e. _V
2	1	pwex 4217	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4453	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3461	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7311	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3639	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2456	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6496	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2263	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 672	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2260	. . . . . . 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 2370	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3647	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6497	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2263	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 672	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7312	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3647	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2263	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 672	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2260	. . . . 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 4536	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3639	. . 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 6485	. . . . . 6 |- 3o = suc 2o
31		df-suc 4407	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2217	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2263	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3305	. . . 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 2167   u. cun 3155   (/)c0 3451   ~Pcpw 3606   {csn 3623   {cpr 3624   suc csuc 4401   1oc1o 6476   2oc2o 6477   3oc3o 6478

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407  df-1o 6483  df-2o 6484  df-3o 6485

This theorem is referenced by:  onntri35  7320