Theorem sucpw1nel3 7582

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 6685	. . . . . . 7 |- 1o e. _V
2	1	pwex 4315	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4557	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3531	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7577	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3721	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2509	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6691	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2305	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 682	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2302	. . . . . . 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 672	. . . . 5 |- -. suc ~P 1o = 1o
14	13	neir 2423	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3729	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6692	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2305	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 682	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7578	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3729	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2305	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 682	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2302	. . . . 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 672	. . 3 |- -. suc ~P 1o = 2o
26	2	sucex 4641	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3721	. . 3 |- ( suc ~P 1o e. { 2o } <-> suc ~P 1o = 2o )
28	25, 27	mtbir 682	. 2 |- -. suc ~P 1o e. { 2o }
29		ioran 764	. . 3 |- ( -. ( suc ~P 1o e. 2o \/ suc ~P 1o e. { 2o } ) <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
30		df-3o 6679	. . . . . 6 |- 3o = suc 2o
31		df-suc 4511	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2259	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2305	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3370	. . . 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 692	. 2 |- ( -. suc ~P 1o e. 3o <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
37	18, 28, 36	mpbir2an 955	1 |- -. suc ~P 1o e. 3o

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   u. cun 3218   (/)c0 3520   ~Pcpw 3685   {csn 3705   {cpr 3706   suc csuc 4505   1oc1o 6670   2oc2o 6671   3oc3o 6672

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  df-2o 6678  df-3o 6679

This theorem is referenced by:  onntri35  7586