Theorem exmidel 4238

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 4229	. . 3 |- (EXMID -> DECID x e. y )
2	1	alrimivv 1889	. 2 |- (EXMID -> A. x A. yDECID x e. y )
3		0ex 4160	. . . 4 |- (/) e. _V
4		eleq1 2259	. . . . . 6 |- ( x = (/) -> ( x e. y <-> (/) e. y ) )
5	4	dcbid 839	. . . . 5 |- ( x = (/) -> (DECID x e. y <-> DECID (/) e. y ) )
6	5	albidv 1838	. . . 4 |- ( x = (/) -> ( A. yDECID x e. y <-> A. yDECID (/) e. y ) )
7	3, 6	spcv 2858	. . 3 |- ( A. x A. yDECID x e. y -> A. yDECID (/) e. y )
8		exmid0el 4237	. . 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 835   A.wal 1362   = wceq 1364   e. wcel 2167   (/)c0 3450  EXMIDwem 4227

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-exmid 4228

This theorem is referenced by: (None)