Theorem pw1nel3 7541

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 7538	. . . . 5 |- ~P 1o =/= (/)
2		pw1ne1 7539	. . . . 5 |- ~P 1o =/= 1o
3	1, 2	nelpri 3713	. . . 4 |- -. ~P 1o e. { (/) , 1o }
4	3	a1i 9	. . 3 |- ( -. EXMID -> -. ~P 1o e. { (/) , 1o } )
5		df2o3 6662	. . . 4 |- 2o = { (/) , 1o }
6	5	eleq2i 2299	. . 3 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
7	4, 6	sylnibr 684	. 2 |- ( -. EXMID -> -. ~P 1o e. 2o )
8		exmidpweq 7169	. . . 4 |- (EXMID <-> ~P 1o = 2o )
9	8	notbii 674	. . 3 |- ( -. EXMID <-> -. ~P 1o = 2o )
10		1oex 6655	. . . . . 6 |- 1o e. _V
11	10	pwex 4296	. . . . 5 |- ~P 1o e. _V
12	11	elsn 3705	. . . 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 6649	. . . . . . 7 |- 3o = suc 2o
16		df-suc 4492	. . . . . . 7 |- suc 2o = ( 2o u. { 2o } )
17	15, 16	eqtri 2253	. . . . . 6 |- 3o = ( 2o u. { 2o } )
18	17	eleq2i 2299	. . . . 5 |- ( ~P 1o e. 3o <-> ~P 1o e. ( 2o u. { 2o } ) )
19		elun 3360	. . . . 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 2203   u. cun 3209   (/)c0 3508   ~Pcpw 3669   {csn 3689   {cpr 3690  EXMIDwem 4307   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-dc 843  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-exmid 4308  df-iord 4487  df-on 4489  df-suc 4492  df-1o 6647  df-2o 6648  df-3o 6649

This theorem is referenced by:  sucpw1ne3  7542  sucpw1nss3  7545