Theorem sucpw1nel3 7556

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 6668	. . . . . . 7 |- 1o e. _V
2	1	pwex 4301	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4543	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3522	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7551	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3710	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2503	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6674	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2301	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 678	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2298	. . . . . . 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 668	. . . . 5 |- -. suc ~P 1o = 1o
14	13	neir 2417	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3718	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6675	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2301	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 678	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7552	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3718	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2301	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 678	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2298	. . . . 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 668	. . 3 |- -. suc ~P 1o = 2o
26	2	sucex 4626	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3710	. . 3 |- ( suc ~P 1o e. { 2o } <-> suc ~P 1o = 2o )
28	25, 27	mtbir 678	. 2 |- -. suc ~P 1o e. { 2o }
29		ioran 760	. . 3 |- ( -. ( suc ~P 1o e. 2o \/ suc ~P 1o e. { 2o } ) <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
30		df-3o 6662	. . . . . 6 |- 3o = suc 2o
31		df-suc 4497	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2255	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2301	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3364	. . . 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 688	. 2 |- ( -. suc ~P 1o e. 3o <-> ( -. suc ~P 1o e. 2o /\ -. suc ~P 1o e. { 2o } ) )
37	18, 28, 36	mpbir2an 951	1 |- -. suc ~P 1o e. 3o

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   u. cun 3212   (/)c0 3512   ~Pcpw 3674   {csn 3694   {cpr 3695   suc csuc 4491   1oc1o 6653   2oc2o 6654   3oc3o 6655

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660  df-2o 6661  df-3o 6662

This theorem is referenced by:  onntri35  7560