Theorem sucpw1nel3 7418

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 6570	. . . . . . 7 |- 1o e. _V
2	1	pwex 4267	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4508	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3501	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7413	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3682	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2489	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6575	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2296	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 675	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2293	. . . . . . 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 666	. . . . 5 |- -. suc ~P 1o = 1o
14	13	neir 2403	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3690	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6576	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2296	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 675	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7414	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3690	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2296	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 675	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2293	. . . . 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 666	. . 3 |- -. suc ~P 1o = 2o
26	2	sucex 4591	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3682	. . 3 |- ( suc ~P 1o e. { 2o } <-> suc ~P 1o = 2o )
28	25, 27	mtbir 675	. 2 |- -. suc ~P 1o e. { 2o }
29		ioran 757	. . 3 |- ( -. ( suc ~P 1o e. 2o \/ suc ~P 1o e. { 2o } ) <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
30		df-3o 6564	. . . . . 6 |- 3o = suc 2o
31		df-suc 4462	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2250	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2296	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3345	. . . 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 685	. 2 |- ( -. suc ~P 1o e. 3o <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
37	18, 28, 36	mpbir2an 948	1 |- -. suc ~P 1o e. 3o

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   u. cun 3195   (/)c0 3491   ~Pcpw 3649   {csn 3666   {cpr 3667   suc csuc 4456   1oc1o 6555   2oc2o 6556   3oc3o 6557

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  df-2o 6563  df-3o 6564

This theorem is referenced by:  onntri35  7422