Theorem exmidn0m 4062

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 694	. . . 4 |- ( (EXMID /\ E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
3		notm0 3330	. . . . . . 7 |- ( -. E. y y e. x <-> x = (/) )
4	3	biimpi 119	. . . . . 6 |- ( -. E. y y e. x -> x = (/) )
5	4	adantl 273	. . . . 5 |- ( (EXMID /\ -. E. y y e. x ) -> x = (/) )
6	5	orcd 693	. . . 4 |- ( (EXMID /\ -. E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
7		exmidexmid 4060	. . . . 5 |- (EXMID -> DECID E. y y e. x )
8		exmiddc 788	. . . . 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 753	. . 3 |- (EXMID -> ( x = (/) \/ E. y y e. x ) )
11	10	alrimiv 1813	. 2 |- (EXMID -> A. x ( x = (/) \/ E. y y e. x ) )
12		orc 674	. . . . . 6 |- ( x = (/) -> ( x = (/) \/ x = { (/) } ) )
13	12	a1d 22	. . . . 5 |- ( x = (/) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		sssnm 3628	. . . . . . . 8 |- ( E. y y e. x -> ( x C_ { (/) } <-> x = { (/) } ) )
15	14	biimpa 292	. . . . . . 7 |- ( ( E. y y e. x /\ x C_ { (/) } ) -> x = { (/) } )
16	15	olcd 694	. . . . . 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 677	. . . 4 |- ( ( x = (/) \/ E. y y e. x ) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
19	18	alimi 1399	. . 3 |- ( A. x ( x = (/) \/ E. y y e. x ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
20		exmid01 4061	. . 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 670  DECID wdc 786   A.wal 1297   = wceq 1299   E.wex 1436   C_ wss 3021   (/)c0 3310   {csn 3474  EXMIDwem 4058

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038

This theorem depends on definitions:  df-bi 116  df-dc 787  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-v 2643  df-dif 3023  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-exmid 4059

This theorem is referenced by:  exmidsssn  4063