Theorem exmidel 4184

Description: Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmidel	|- (EXMID <-> A. x A. yDECID x e. y )

Distinct variable group:   x, y

Proof of Theorem exmidel

Step	Hyp	Ref	Expression
1		exmidexmid 4175	. . 3 |- (EXMID -> DECID x e. y )
2	1	alrimivv 1863	. 2 |- (EXMID -> A. x A. yDECID x e. y )
3		0ex 4109	. . . 4 |- (/) e. _V
4		eleq1 2229	. . . . . 6 |- ( x = (/) -> ( x e. y <-> (/) e. y ) )
5	4	dcbid 828	. . . . 5 |- ( x = (/) -> (DECID x e. y <-> DECID (/) e. y ) )
6	5	albidv 1812	. . . 4 |- ( x = (/) -> ( A. yDECID x e. y <-> A. yDECID (/) e. y ) )
7	3, 6	spcv 2820	. . 3 |- ( A. x A. yDECID x e. y -> A. yDECID (/) e. y )
8		exmid0el 4183	. . 3 |- (EXMID <-> A. yDECID (/) e. y )
9	7, 8	sylibr 133	. 2 |- ( A. x A. yDECID x e. y -> EXMID )
10	2, 9	impbii 125	1 |- (EXMID <-> A. x A. yDECID x e. y )

Syntax hints:   <-> wb 104  DECID wdc 824   A.wal 1341   = wceq 1343   e. wcel 2136   (/)c0 3409  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: (None)