Theorem pw1nel3 7580

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 7577	. . . . 5 |- ~P 1o =/= (/)
2		pw1ne1 7578	. . . . 5 |- ~P 1o =/= 1o
3	1, 2	nelpri 3729	. . . 4 |- -. ~P 1o e. { (/) , 1o }
4	3	a1i 9	. . 3 |- ( -. EXMID -> -. ~P 1o e. { (/) , 1o } )
5		df2o3 6692	. . . 4 |- 2o = { (/) , 1o }
6	5	eleq2i 2305	. . 3 |- ( ~P 1o e. 2o <-> ~P 1o e. { (/) , 1o } )
7	4, 6	sylnibr 688	. 2 |- ( -. EXMID -> -. ~P 1o e. 2o )
8		exmidpweq 7206	. . . 4 |- (EXMID <-> ~P 1o = 2o )
9	8	notbii 678	. . 3 |- ( -. EXMID <-> -. ~P 1o = 2o )
10		1oex 6685	. . . . . 6 |- 1o e. _V
11	10	pwex 4315	. . . . 5 |- ~P 1o e. _V
12	11	elsn 3721	. . . 4 |- ( ~P 1o e. { 2o } <-> ~P 1o = 2o )
13	12	notbii 678	. . 3 |- ( -. ~P 1o e. { 2o } <-> -. ~P 1o = 2o )
14	9, 13	sylbb2 138	. 2 |- ( -. EXMID -> -. ~P 1o e. { 2o } )
15		df-3o 6679	. . . . . . 7 |- 3o = suc 2o
16		df-suc 4511	. . . . . . 7 |- suc 2o = ( 2o u. { 2o } )
17	15, 16	eqtri 2259	. . . . . 6 |- 3o = ( 2o u. { 2o } )
18	17	eleq2i 2305	. . . . 5 |- ( ~P 1o e. 3o <-> ~P 1o e. ( 2o u. { 2o } ) )
19		elun 3370	. . . . 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 678	. . 3 |- ( -. ~P 1o e. 3o <-> -. ( ~P 1o e. 2o \/ ~P 1o e. { 2o } ) )
22		ioran 764	. . 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 421	1 |- ( -. EXMID -> -. ~P 1o e. 3o )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   u. cun 3218   (/)c0 3520   ~Pcpw 3685   {csn 3705   {cpr 3706  EXMIDwem 4326   suc csuc 4505   1oc1o 6670   2oc2o 6671   3oc3o 6672

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508  df-suc 4511  df-1o 6677  df-2o 6678  df-3o 6679

This theorem is referenced by:  sucpw1ne3  7581  sucpw1nss3  7584