Theorem sucpw1nel3 7347

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 6512	. . . . . . 7 |- 1o e. _V
2	1	pwex 4228	. . . . . 6 |- ~P 1o e. _V
3	2	sucid 4465	. . . . 5 |- ~P 1o e. suc ~P 1o
4	3	ne0ii 3470	. . . 4 |- suc ~P 1o =/= (/)
5		pw1ne0 7342	. . . . . . . 8 |- ~P 1o =/= (/)
6	2	elsn 3649	. . . . . . . 8 |- ( ~P 1o e. { (/) } <-> ~P 1o = (/) )
7	5, 6	nemtbir 2465	. . . . . . 7 |- -. ~P 1o e. { (/) }
8		df1o2 6517	. . . . . . . 8 |- 1o = { (/) }
9	8	eleq2i 2272	. . . . . . 7 |- ( ~P 1o e. 1o <-> ~P 1o e. { (/) } )
10	7, 9	mtbir 673	. . . . . 6 |- -. ~P 1o e. 1o
11		eleq2 2269	. . . . . . 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 2379	. . . 4 |- suc ~P 1o =/= 1o
15	4, 14	nelpri 3657	. . 3 |- -. suc ~P 1o e. { (/) , 1o }
16		df2o3 6518	. . . 4 |- 2o = { (/) , 1o }
17	16	eleq2i 2272	. . 3 |- ( suc ~P 1o e. 2o <-> suc ~P 1o e. { (/) , 1o } )
18	15, 17	mtbir 673	. 2 |- -. suc ~P 1o e. 2o
19		pw1ne1 7343	. . . . . 6 |- ~P 1o =/= 1o
20	5, 19	nelpri 3657	. . . . 5 |- -. ~P 1o e. { (/) , 1o }
21	16	eleq2i 2272	. . . . 5 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
22	20, 21	mtbir 673	. . . 4 |- -. ~P 1o e. 2o
23		eleq2 2269	. . . . 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 4548	. . . 4 |- suc ~P 1o e. _V
27	26	elsn 3649	. . 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 6506	. . . . . 6 |- 3o = suc 2o
31		df-suc 4419	. . . . . 6 |- suc 2o = ( 2o u. { 2o } )
32	30, 31	eqtri 2226	. . . . 5 |- 3o = ( 2o u. { 2o } )
33	32	eleq2i 2272	. . . 4 |- ( suc ~P 1o e. 3o <-> suc ~P 1o e. ( 2o u. { 2o } ) )
34		elun 3314	. . . 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 2176   u. cun 3164   (/)c0 3460   ~Pcpw 3616   {csn 3633   {cpr 3634   suc csuc 4413   1oc1o 6497   2oc2o 6498   3oc3o 6499

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4144  df-iord 4414  df-on 4416  df-suc 4419  df-1o 6504  df-2o 6505  df-3o 6506

This theorem is referenced by:  onntri35  7351