Theorem exmidpweq 6887

Description: Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)

Assertion

Ref	Expression
exmidpweq	|- (EXMID <-> ~P 1o = 2o )

Proof of Theorem exmidpweq

Step	Hyp	Ref	Expression
1		exmid01 4184	. . . . . . . 8 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2	1	biimpi 119	. . . . . . 7 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
3	2	19.21bi 1551	. . . . . 6 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
4		df1o2 6408	. . . . . . . . 9 |- 1o = { (/) }
5	4	pweqi 3570	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	5	eleq2i 2237	. . . . . . 7 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
7		velpw 3573	. . . . . . 7 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
8	6, 7	bitri 183	. . . . . 6 |- ( x e. ~P 1o <-> x C_ { (/) } )
9		vex 2733	. . . . . . 7 |- x e. _V
10	9	elpr 3604	. . . . . 6 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
11	3, 8, 10	3imtr4g 204	. . . . 5 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
12	11	ssrdv 3153	. . . 4 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
13		pwpw0ss 3791	. . . . . 6 |- { (/) , { (/) } } C_ ~P { (/) }
14	13, 5	sseqtrri 3182	. . . . 5 |- { (/) , { (/) } } C_ ~P 1o
15	14	a1i 9	. . . 4 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
16	12, 15	eqssd 3164	. . 3 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
17		df2o2 6410	. . 3 |- 2o = { (/) , { (/) } }
18	16, 17	eqtr4di 2221	. 2 |- (EXMID -> ~P 1o = 2o )
19		simpr 109	. . . . . . . . 9 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x C_ { (/) } )
20	19, 7	sylibr 133	. . . . . . . 8 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. ~P { (/) } )
21	20, 5	eleqtrrdi 2264	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. ~P 1o )
22		simpl 108	. . . . . . . 8 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = 2o )
23	22, 17	eqtrdi 2219	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = { (/) , { (/) } } )
24	21, 23	eleqtrd 2249	. . . . . 6 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> x e. { (/) , { (/) } } )
25	24, 10	sylib 121	. . . . 5 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
26	25	ex 114	. . . 4 |- ( ~P 1o = 2o -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
27	26	alrimiv 1867	. . 3 |- ( ~P 1o = 2o -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
28	27, 1	sylibr 133	. 2 |- ( ~P 1o = 2o -> EXMID )
29	18, 28	impbii 125	1 |- (EXMID <-> ~P 1o = 2o )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   A.wal 1346   = wceq 1348   e. wcel 2141   C_ wss 3121   (/)c0 3414   ~Pcpw 3566   {csn 3583   {cpr 3584  EXMIDwem 4180   1oc1o 6388   2oc2o 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-exmid 4181  df-suc 4356  df-1o 6395  df-2o 6396

This theorem is referenced by:  pw1fin  6888  pw1nel3  7208  3nsssucpw1  7213  onntri35  7214