Theorem exmidel 4123

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 4115	. . 3 |- (EXMID -> DECID x e. y )
2	1	alrimivv 1847	. 2 |- (EXMID -> A. x A. yDECID x e. y )
3		0ex 4050	. . . 4 |- (/) e. _V
4		eleq1 2200	. . . . . 6 |- ( x = (/) -> ( x e. y <-> (/) e. y ) )
5	4	dcbid 823	. . . . 5 |- ( x = (/) -> (DECID x e. y <-> DECID (/) e. y ) )
6	5	albidv 1796	. . . 4 |- ( x = (/) -> ( A. yDECID x e. y <-> A. yDECID (/) e. y ) )
7	3, 6	spcv 2774	. . 3 |- ( A. x A. yDECID x e. y -> A. yDECID (/) e. y )
8		exmid0el 4122	. . 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 819   A.wal 1329   = wceq 1331   e. wcel 1480   (/)c0 3358  EXMIDwem 4113

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-exmid 4114

This theorem is referenced by: (None)