Theorem sucpw1nel3 7379

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 6533	. . . . . . 7 |- 1o e. _V
2	1	pwex 4243	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4482	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3478	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7374	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3659	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2467	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6538	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2274	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 673	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2271	. . . . . . 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 664	. . . . 5 |- -. suc ~P 1o = 1o
14	13	neir 2381	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3667	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6539	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2274	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 673	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7375	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3667	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2274	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 673	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2271	. . . . 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 664	. . 3 |- -. suc ~P 1o = 2o
26	2	sucex 4565	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3659	. . 3 |- ( suc ~P 1o e. { 2o } <-> suc ~P 1o = 2o )
28	25, 27	mtbir 673	. 2 |- -. suc ~P 1o e. { 2o }
29		ioran 754	. . 3 |- ( -. ( suc ~P 1o e. 2o \/ suc ~P 1o e. { 2o } ) <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
30		df-3o 6527	. . . . . 6 |- 3o = suc 2o
31		df-suc 4436	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2228	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2274	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3322	. . . 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 683	. 2 |- ( -. suc ~P 1o e. 3o <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
37	18, 28, 36	mpbir2an 945	1 |- -. suc ~P 1o e. 3o

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2178   u. cun 3172   (/)c0 3468   ~Pcpw 3626   {csn 3643   {cpr 3644   suc csuc 4430   1oc1o 6518   2oc2o 6519   3oc3o 6520

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436  df-1o 6525  df-2o 6526  df-3o 6527

This theorem is referenced by:  onntri35  7383