Theorem exmid0el 4300

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 4292	. . 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 1504	. . 3 |- ( A. xDECID (/) e. x -> A. x ( x C_ { (/) } -> DECID (/) e. x ) )
5		df-exmid 4291	. . 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 842   A.wal 1396   e. wcel 2202   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-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:  exmidel  4301