Theorem exmid0el 4190

Description: Excluded middle is equivalent to decidability of (/) being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmid0el	|- (EXMID <-> A. xDECID (/) e. x )

Proof of Theorem exmid0el

Step	Hyp	Ref	Expression
1		exmidexmid 4182	. . 3 |- (EXMID -> DECID (/) e. x )
2	1	alrimiv 1867	. 2 |- (EXMID -> A. xDECID (/) e. x )
3		ax-1 6	. . . 4 |- (DECID (/) e. x -> ( x C_ { (/) } -> DECID (/) e. x ) )
4	3	alimi 1448	. . 3 |- ( A. xDECID (/) e. x -> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
5		df-exmid 4181	. . 3 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
6	4, 5	sylibr 133	. 2 |- ( A. xDECID (/) e. x -> EXMID )
7	2, 6	impbii 125	1 |- (EXMID <-> A. xDECID (/) e. x )

Syntax hints:   -> wi 4   <-> wb 104  DECID wdc 829   A.wal 1346   e. wcel 2141   C_ wss 3121   (/)c0 3414   {csn 3583  EXMIDwem 4180

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-exmid 4181

This theorem is referenced by:  exmidel  4191