Theorem exmidel 4295

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 4286	. . 3 |- (EXMID -> DECID x e. y )
2	1	alrimivv 1923	. 2 |- (EXMID -> A. x A. yDECID x e. y )
3		0ex 4216	. . . 4 |- (/) e. _V
4		eleq1 2294	. . . . . 6 |- ( x = (/) -> ( x e. y <-> (/) e. y ) )
5	4	dcbid 845	. . . . 5 |- ( x = (/) -> (DECID x e. y <-> DECID (/) e. y ) )
6	5	albidv 1872	. . . 4 |- ( x = (/) -> ( A. yDECID x e. y <-> A. yDECID (/) e. y ) )
7	3, 6	spcv 2900	. . 3 |- ( A. x A. yDECID x e. y -> A. yDECID (/) e. y )
8		exmid0el 4294	. . 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 841   A.wal 1395   = wceq 1397   e. wcel 2202   (/)c0 3494  EXMIDwem 4284

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-exmid 4285

This theorem is referenced by: (None)