Theorem exmidel 4265

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 4256	. . 3 |- (EXMID -> DECID x e. y )
2	1	alrimivv 1899	. 2 |- (EXMID -> A. x A. yDECID x e. y )
3		0ex 4187	. . . 4 |- (/) e. _V
4		eleq1 2270	. . . . . 6 |- ( x = (/) -> ( x e. y <-> (/) e. y ) )
5	4	dcbid 840	. . . . 5 |- ( x = (/) -> (DECID x e. y <-> DECID (/) e. y ) )
6	5	albidv 1848	. . . 4 |- ( x = (/) -> ( A. yDECID x e. y <-> A. yDECID (/) e. y ) )
7	3, 6	spcv 2874	. . 3 |- ( A. x A. yDECID x e. y -> A. yDECID (/) e. y )
8		exmid0el 4264	. . 3 |- (EXMID <-> A. yDECID (/) e. y )
9	7, 8	sylibr 134	. 2 |- ( A. x A. yDECID x e. y -> EXMID )
10	2, 9	impbii 126	1 |- (EXMID <-> A. x A. yDECID x e. y )

Syntax hints:   <-> wb 105  DECID wdc 836   A.wal 1371   = wceq 1373   e. wcel 2178   (/)c0 3468  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: (None)