Theorem exmidel 4249

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 4240	. . 3 |- (EXMID -> DECID x e. y )
2	1	alrimivv 1898	. 2 |- (EXMID -> A. x A. yDECID x e. y )
3		0ex 4171	. . . 4 |- (/) e. _V
4		eleq1 2268	. . . . . 6 |- ( x = (/) -> ( x e. y <-> (/) e. y ) )
5	4	dcbid 840	. . . . 5 |- ( x = (/) -> (DECID x e. y <-> DECID (/) e. y ) )
6	5	albidv 1847	. . . 4 |- ( x = (/) -> ( A. yDECID x e. y <-> A. yDECID (/) e. y ) )
7	3, 6	spcv 2867	. . 3 |- ( A. x A. yDECID x e. y -> A. yDECID (/) e. y )
8		exmid0el 4248	. . 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 2176   (/)c0 3460  EXMIDwem 4238

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-exmid 4239

This theorem is referenced by: (None)