Theorem exmidn0m 4119

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 109	. . . . 5 |- ( (EXMID /\ E. y y e. x ) -> E. y y e. x )
2	1	olcd 723	. . . 4 |- ( (EXMID /\ E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
3		notm0 3378	. . . . . . 7 |- ( -. E. y y e. x <-> x = (/) )
4	3	biimpi 119	. . . . . 6 |- ( -. E. y y e. x -> x = (/) )
5	4	adantl 275	. . . . 5 |- ( (EXMID /\ -. E. y y e. x ) -> x = (/) )
6	5	orcd 722	. . . 4 |- ( (EXMID /\ -. E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
7		exmidexmid 4115	. . . . 5 |- (EXMID -> DECID E. y y e. x )
8		exmiddc 821	. . . . 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 787	. . 3 |- (EXMID -> ( x = (/) \/ E. y y e. x ) )
11	10	alrimiv 1846	. 2 |- (EXMID -> A. x ( x = (/) \/ E. y y e. x ) )
12		orc 701	. . . . . 6 |- ( x = (/) -> ( x = (/) \/ x = { (/) } ) )
13	12	a1d 22	. . . . 5 |- ( x = (/) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		sssnm 3676	. . . . . . . 8 |- ( E. y y e. x -> ( x C_ { (/) } <-> x = { (/) } ) )
15	14	biimpa 294	. . . . . . 7 |- ( ( E. y y e. x /\ x C_ { (/) } ) -> x = { (/) } )
16	15	olcd 723	. . . . . 6 |- ( ( E. y y e. x /\ x C_ { (/) } ) -> ( x = (/) \/ x = { (/) } ) )
17	16	ex 114	. . . . 5 |- ( E. y y e. x -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
18	13, 17	jaoi 705	. . . 4 |- ( ( x = (/) \/ E. y y e. x ) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
19	18	alimi 1431	. . 3 |- ( A. x ( x = (/) \/ E. y y e. x ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
20		exmid01 4116	. . 3 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
21	19, 20	sylibr 133	. 2 |- ( A. x ( x = (/) \/ E. y y e. x ) -> EXMID )
22	11, 21	impbii 125	1 |- (EXMID <-> A. x ( x = (/) \/ E. y y e. x ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   A.wal 1329   = wceq 1331   E.wex 1468   C_ wss 3066   (/)c0 3358   {csn 3522  EXMIDwem 4113

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-exmid 4114

This theorem is referenced by:  exmidsssn  4120