Theorem exmidn0m 4216

Description: Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)

Assertion

Ref	Expression
exmidn0m	|- (EXMID <-> A. x ( x = (/) \/ E. y y e. x ) )

Distinct variable group:   x, y

Proof of Theorem exmidn0m

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( (EXMID /\ E. y y e. x ) -> E. y y e. x )
2	1	olcd 735	. . . 4 |- ( (EXMID /\ E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
3		notm0 3458	. . . . . . 7 |- ( -. E. y y e. x <-> x = (/) )
4	3	biimpi 120	. . . . . 6 |- ( -. E. y y e. x -> x = (/) )
5	4	adantl 277	. . . . 5 |- ( (EXMID /\ -. E. y y e. x ) -> x = (/) )
6	5	orcd 734	. . . 4 |- ( (EXMID /\ -. E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
7		exmidexmid 4211	. . . . 5 |- (EXMID -> DECID E. y y e. x )
8		exmiddc 837	. . . . 5 |- (DECID E. y y e. x -> ( E. y y e. x \/ -. E. y y e. x ) )
9	7, 8	syl 14	. . . 4 |- (EXMID -> ( E. y y e. x \/ -. E. y y e. x ) )
10	2, 6, 9	mpjaodan 799	. . 3 |- (EXMID -> ( x = (/) \/ E. y y e. x ) )
11	10	alrimiv 1885	. 2 |- (EXMID -> A. x ( x = (/) \/ E. y y e. x ) )
12		orc 713	. . . . . 6 |- ( x = (/) -> ( x = (/) \/ x = { (/) } ) )
13	12	a1d 22	. . . . 5 |- ( x = (/) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		sssnm 3769	. . . . . . . 8 |- ( E. y y e. x -> ( x C_ { (/) } <-> x = { (/) } ) )
15	14	biimpa 296	. . . . . . 7 |- ( ( E. y y e. x /\ x C_ { (/) } ) -> x = { (/) } )
16	15	olcd 735	. . . . . 6 |- ( ( E. y y e. x /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
17	16	ex 115	. . . . 5 |- ( E. y y e. x -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
18	13, 17	jaoi 717	. . . 4 |- ( ( x = (/) \/ E. y y e. x ) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
19	18	alimi 1466	. . 3 |- ( A. x ( x = (/) \/ E. y y e. x ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
20		exmid01 4213	. . 3 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
21	19, 20	sylibr 134	. 2 |- ( A. x ( x = (/) \/ E. y y e. x ) -> EXMID )
22	11, 21	impbii 126	1 |- (EXMID <-> A. x ( x = (/) \/ E. y y e. x ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   A.wal 1362   = wceq 1364   E.wex 1503   C_ wss 3144   (/)c0 3437   {csn 3607  EXMIDwem 4209

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-exmid 4210

This theorem is referenced by:  exmidsssn  4217