Theorem pw1nel3 7416

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 7413	. . . . 5 |- ~P 1o =/= (/)
2		pw1ne1 7414	. . . . 5 |- ~P 1o =/= 1o
3	1, 2	nelpri 3690	. . . 4 |- -. ~P 1o e. { (/) , 1o }
4	3	a1i 9	. . 3 |- ( -. EXMID -> -. ~P 1o e. { (/) , 1o } )
5		df2o3 6576	. . . 4 |- 2o = { (/) , 1o }
6	5	eleq2i 2296	. . 3 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
7	4, 6	sylnibr 681	. 2 |- ( -. EXMID -> -. ~P 1o e. 2o )
8		exmidpweq 7071	. . . 4 |- (EXMID <-> ~P 1o = 2o )
9	8	notbii 672	. . 3 |- ( -. EXMID <-> -. ~P 1o = 2o )
10		1oex 6570	. . . . . 6 |- 1o e. _V
11	10	pwex 4267	. . . . 5 |- ~P 1o e. _V
12	11	elsn 3682	. . . 4 |- ( ~P 1o e. { 2o } <-> ~P 1o = 2o )
13	12	notbii 672	. . 3 |- ( -. ~P 1o e. { 2o } <-> -. ~P 1o = 2o )
14	9, 13	sylbb2 138	. 2 |- ( -. EXMID -> -. ~P 1o e. { 2o } )
15		df-3o 6564	. . . . . . 7 |- 3o = suc 2o
16		df-suc 4462	. . . . . . 7 |- suc 2o = ( 2o u. { 2o } )
17	15, 16	eqtri 2250	. . . . . 6 |- 3o = ( 2o u. { 2o } )
18	17	eleq2i 2296	. . . . 5 |- ( ~P 1o e. 3o <-> ~P 1o e. ( 2o u. { 2o } ) )
19		elun 3345	. . . . 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 672	. . 3 |- ( -. ~P 1o e. 3o <-> -. ( ~P 1o e. 2o \/ ~P 1o e. { 2o } ) )
22		ioran 757	. . 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 713   = wceq 1395   e. wcel 2200   u. cun 3195   (/)c0 3491   ~Pcpw 3649   {csn 3666   {cpr 3667  EXMIDwem 4278   suc csuc 4456   1oc1o 6555   2oc2o 6556   3oc3o 6557

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-exmid 4279  df-iord 4457  df-on 4459  df-suc 4462  df-1o 6562  df-2o 6563  df-3o 6564

This theorem is referenced by:  sucpw1ne3  7417  sucpw1nss3  7420