Theorem exmidn0m 4180

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 724	. . . 4 |- ( (EXMID /\ E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
3		notm0 3429	. . . . . . 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 723	. . . 4 |- ( (EXMID /\ -. E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
7		exmidexmid 4175	. . . . 5 |- (EXMID -> DECID E. y y e. x )
8		exmiddc 826	. . . . 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 788	. . 3 |- (EXMID -> ( x = (/) \/ E. y y e. x ) )
11	10	alrimiv 1862	. 2 |- (EXMID -> A. x ( x = (/) \/ E. y y e. x ) )
12		orc 702	. . . . . 6 |- ( x = (/) -> ( x = (/) \/ x = { (/) } ) )
13	12	a1d 22	. . . . 5 |- ( x = (/) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		sssnm 3734	. . . . . . . 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 724	. . . . . 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 706	. . . 4 |- ( ( x = (/) \/ E. y y e. x ) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
19	18	alimi 1443	. . 3 |- ( A. x ( x = (/) \/ E. y y e. x ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
20		exmid01 4177	. . 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 698  DECID wdc 824   A.wal 1341   = wceq 1343   E.wex 1480   C_ wss 3116   (/)c0 3409   {csn 3576  EXMIDwem 4173

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-exmid 4174

This theorem is referenced by:  exmidsssn  4181