Theorem exmidn0m 4297

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 742	. . . 4 |- ( (EXMID /\ E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
3		notm0 3517	. . . . . . 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 741	. . . 4 |- ( (EXMID /\ -. E. y y e. x ) -> ( x = (/) \/ E. y y e. x ) )
7		exmidexmid 4292	. . . . 5 |- (EXMID -> DECID E. y y e. x )
8		exmiddc 844	. . . . 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 806	. . 3 |- (EXMID -> ( x = (/) \/ E. y y e. x ) )
11	10	alrimiv 1922	. 2 |- (EXMID -> A. x ( x = (/) \/ E. y y e. x ) )
12		orc 720	. . . . . 6 |- ( x = (/) -> ( x = (/) \/ x = { (/) } ) )
13	12	a1d 22	. . . . 5 |- ( x = (/) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
14		sssnm 3842	. . . . . . . 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 742	. . . . . 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 724	. . . 4 |- ( ( x = (/) \/ E. y y e. x ) -> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
19	18	alimi 1504	. . 3 |- ( A. x ( x = (/) \/ E. y y e. x ) -> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
20		exmid01 4294	. . 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 716  DECID wdc 842   A.wal 1396   = wceq 1398   E.wex 1541   C_ wss 3201   (/)c0 3496   {csn 3673  EXMIDwem 4290

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-exmid 4291

This theorem is referenced by:  exmidsssn  4298