Theorem exmid0el 4264

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 4256	. . 3 |- (EXMID -> DECID (/) e. x )
2	1	alrimiv 1898	. 2 |- (EXMID -> A. xDECID (/) e. x )
3		ax-1 6	. . . 4 |- (DECID (/) e. x -> ( x C_ { (/) } -> DECID (/) e. x ) )
4	3	alimi 1479	. . 3 |- ( A. xDECID (/) e. x -> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
5		df-exmid 4255	. . 3 |- (EXMID <-> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
6	4, 5	sylibr 134	. 2 |- ( A. xDECID (/) e. x -> EXMID )
7	2, 6	impbii 126	1 |- (EXMID <-> A. xDECID (/) e. x )

Syntax hints:   -> wi 4   <-> wb 105  DECID wdc 836   A.wal 1371   e. wcel 2178   C_ wss 3174   (/)c0 3468   {csn 3643  EXMIDwem 4254

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-exmid 4255

This theorem is referenced by:  exmidel  4265