Theorem pw1nel3 7509

Description: Negated excluded middle implies that 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
pw1nel3	|- ( -. EXMID -> -. ~P 1o e. 3o )

Proof of Theorem pw1nel3

Step	Hyp	Ref	Expression
1		pw1ne0 7506	. . . . 5 |- ~P 1o =/= (/)
2		pw1ne1 7507	. . . . 5 |- ~P 1o =/= 1o
3	1, 2	nelpri 3697	. . . 4 |- -. ~P 1o e. { (/) , 1o }
4	3	a1i 9	. . 3 |- ( -. EXMID -> -. ~P 1o e. { (/) , 1o } )
5		df2o3 6640	. . . 4 |- 2o = { (/) , 1o }
6	5	eleq2i 2298	. . 3 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
7	4, 6	sylnibr 684	. 2 |- ( -. EXMID -> -. ~P 1o e. 2o )
8		exmidpweq 7144	. . . 4 |- (EXMID <-> ~P 1o = 2o )
9	8	notbii 674	. . 3 |- ( -. EXMID <-> -. ~P 1o = 2o )
10		1oex 6633	. . . . . 6 |- 1o e. _V
11	10	pwex 4279	. . . . 5 |- ~P 1o e. _V
12	11	elsn 3689	. . . 4 |- ( ~P 1o e. { 2o } <-> ~P 1o = 2o )
13	12	notbii 674	. . 3 |- ( -. ~P 1o e. { 2o } <-> -. ~P 1o = 2o )
14	9, 13	sylbb2 138	. 2 |- ( -. EXMID -> -. ~P 1o e. { 2o } )
15		df-3o 6627	. . . . . . 7 |- 3o = suc 2o
16		df-suc 4474	. . . . . . 7 |- suc 2o = ( 2o u. { 2o } )
17	15, 16	eqtri 2252	. . . . . 6 |- 3o = ( 2o u. { 2o } )
18	17	eleq2i 2298	. . . . 5 |- ( ~P 1o e. 3o <-> ~P 1o e. ( 2o u. { 2o } ) )
19		elun 3350	. . . . 5 |- ( ~P 1o e. ( 2o u. { 2o } ) <-> ( ~P 1o e. 2o \/ ~P 1o e. { 2o } ) )
20	18, 19	bitri 184	. . . 4 |- ( ~P 1o e. 3o <-> ( ~P 1o e. 2o \/ ~P 1o e. { 2o } ) )
21	20	notbii 674	. . 3 |- ( -. ~P 1o e. 3o <-> -. ( ~P 1o e. 2o \/ ~P 1o e. { 2o } ) )
22		ioran 760	. . 3 |- ( -. ( ~P 1o e. 2o \/ ~P 1o e. { 2o } ) <-> ( -. ~P 1o e. 2o /\ -. ~P 1o e. { 2o } ) )
23	21, 22	bitri 184	. 2 |- ( -. ~P 1o e. 3o <-> ( -. ~P 1o e. 2o /\ -. ~P 1o e. { 2o } ) )
24	7, 14, 23	sylanbrc 417	1 |- ( -. EXMID -> -. ~P 1o e. 3o )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2202   u. cun 3199   (/)c0 3496   ~Pcpw 3656   {csn 3673   {cpr 3674  EXMIDwem 4290   suc csuc 4468   1oc1o 6618   2oc2o 6619   3oc3o 6620

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-exmid 4291  df-iord 4469  df-on 4471  df-suc 4474  df-1o 6625  df-2o 6626  df-3o 6627

This theorem is referenced by:  sucpw1ne3  7510  sucpw1nss3  7513