Theorem exmid0el 4183

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 4175	. . 3 |- (EXMID -> DECID (/) e. x )
2	1	alrimiv 1862	. 2 |- (EXMID -> A. xDECID (/) e. x )
3		ax-1 6	. . . 4 |- (DECID (/) e. x -> ( x C_ { (/) } -> DECID (/) e. x ) )
4	3	alimi 1443	. . 3 |- ( A. xDECID (/) e. x -> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
5		df-exmid 4174	. . 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 824   A.wal 1341   e. wcel 2136   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-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:  exmidel  4184