Theorem exmidpweq 6847

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 4158	. . . . . . . 8 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2	1	biimpi 119	. . . . . . 7 |- (EXMID -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
3	2	19.21bi 1538	. . . . . 6 |- (EXMID -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
4		df1o2 6370	. . . . . . . . 9 |- 1o = { (/) }
5	4	pweqi 3547	. . . . . . . 8 |- ~P 1o = ~P { (/) }
6	5	eleq2i 2224	. . . . . . 7 |- ( x e. ~P 1o <-> x e. ~P { (/) } )
7		velpw 3550	. . . . . . 7 |- ( x e. ~P { (/) } <-> x C_ { (/) } )
8	6, 7	bitri 183	. . . . . 6 |- ( x e. ~P 1o <-> x C_ { (/) } )
9		vex 2715	. . . . . . 7 |- x e. _V
10	9	elpr 3581	. . . . . 6 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
11	3, 8, 10	3imtr4g 204	. . . . 5 |- (EXMID -> ( x e. ~P 1o -> x e. { (/) , { (/) } } ) )
12	11	ssrdv 3134	. . . 4 |- (EXMID -> ~P 1o C_ { (/) , { (/) } } )
13		pwpw0ss 3767	. . . . . 6 |- { (/) , { (/) } } C_ ~P { (/) }
14	13, 5	sseqtrri 3163	. . . . 5 |- { (/) , { (/) } } C_ ~P 1o
15	14	a1i 9	. . . 4 |- (EXMID -> { (/) , { (/) } } C_ ~P 1o )
16	12, 15	eqssd 3145	. . 3 |- (EXMID -> ~P 1o = { (/) , { (/) } } )
17		df2o2 6372	. . 3 |- 2o = { (/) , { (/) } }
18	16, 17	eqtr4di 2208	. 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 2251	. . . . . . 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 2206	. . . . . . 7 |- ( ( ~P 1o = 2o /\ x C_ { (/) } ) -> ~P 1o = { (/) , { (/) } } )
24	21, 23	eleqtrd 2236	. . . . . 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 1854	. . 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 698   A.wal 1333   = wceq 1335   e. wcel 2128   C_ wss 3102   (/)c0 3394   ~Pcpw 3543   {csn 3560   {cpr 3561  EXMIDwem 4154   1oc1o 6350   2oc2o 6351

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-exmid 4155  df-suc 4330  df-1o 6357  df-2o 6358

This theorem is referenced by:  pw1fin  6848  pw1nel3  7149  3nsssucpw1  7154  onntri35  7155