Theorem sucpw1nel3 7543

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 6655	. . . . . . 7 |- 1o e. _V
2	1	pwex 4296	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4538	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3518	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7538	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3705	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2501	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6661	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2299	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 678	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2296	. . . . . . 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 2415	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3713	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6662	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2299	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 678	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7539	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3713	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2299	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 678	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2296	. . . . 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 4621	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3705	. . 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 6649	. . . . . 6 |- 3o = suc 2o
31		df-suc 4492	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2253	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2299	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3360	. . . 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 2203   u. cun 3209   (/)c0 3508   ~Pcpw 3669   {csn 3689   {cpr 3690   suc csuc 4486   1oc1o 6640   2oc2o 6641   3oc3o 6642

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492  df-1o 6647  df-2o 6648  df-3o 6649

This theorem is referenced by:  onntri35  7547