Theorem exmid0el 4294

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 4286	. . 3 |- (EXMID -> DECID (/) e. x )
2	1	alrimiv 1922	. 2 |- (EXMID -> A. xDECID (/) e. x )
3		ax-1 6	. . . 4 |- (DECID (/) e. x -> ( x C_ { (/) } -> DECID (/) e. x ) )
4	3	alimi 1503	. . 3 |- ( A. xDECID (/) e. x -> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
5		df-exmid 4285	. . 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 841   A.wal 1395   e. wcel 2202   C_ wss 3200   (/)c0 3494   {csn 3669  EXMIDwem 4284

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-exmid 4285

This theorem is referenced by:  exmidel  4295