Theorem pw1nel3 7554

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 7551	. . . . 5 |- ~P 1o =/= (/)
2		pw1ne1 7552	. . . . 5 |- ~P 1o =/= 1o
3	1, 2	nelpri 3718	. . . 4 |- -. ~P 1o e. { (/) , 1o }
4	3	a1i 9	. . 3 |- ( -. EXMID -> -. ~P 1o e. { (/) , 1o } )
5		df2o3 6675	. . . 4 |- 2o = { (/) , 1o }
6	5	eleq2i 2301	. . 3 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
7	4, 6	sylnibr 684	. 2 |- ( -. EXMID -> -. ~P 1o e. 2o )
8		exmidpweq 7182	. . . 4 |- (EXMID <-> ~P 1o = 2o )
9	8	notbii 674	. . 3 |- ( -. EXMID <-> -. ~P 1o = 2o )
10		1oex 6668	. . . . . 6 |- 1o e. _V
11	10	pwex 4301	. . . . 5 |- ~P 1o e. _V
12	11	elsn 3710	. . . 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 6662	. . . . . . 7 |- 3o = suc 2o
16		df-suc 4497	. . . . . . 7 |- suc 2o = ( 2o u. { 2o } )
17	15, 16	eqtri 2255	. . . . . 6 |- 3o = ( 2o u. { 2o } )
18	17	eleq2i 2301	. . . . 5 |- ( ~P 1o e. 3o <-> ~P 1o e. ( 2o u. { 2o } ) )
19		elun 3364	. . . . 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 2205   u. cun 3212   (/)c0 3512   ~Pcpw 3674   {csn 3694   {cpr 3695  EXMIDwem 4312   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-dc 843  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-exmid 4313  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660  df-2o 6661  df-3o 6662

This theorem is referenced by:  sucpw1ne3  7555  sucpw1nss3  7558