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Theorem exmidel 4128
 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 DECID
Distinct variable group:   ,

Proof of Theorem exmidel
StepHypRef Expression
1 exmidexmid 4120 . . 3 EXMID DECID
21alrimivv 1847 . 2 EXMID DECID
3 0ex 4055 . . . 4
4 eleq1 2202 . . . . . 6
54dcbid 823 . . . . 5 DECID DECID
65albidv 1796 . . . 4 DECID DECID
73, 6spcv 2779 . . 3 DECID DECID
8 exmid0el 4127 . . 3 EXMID DECID
97, 8sylibr 133 . 2 DECID EXMID
102, 9impbii 125 1 EXMID DECID
 Colors of variables: wff set class Syntax hints:   wb 104  DECID wdc 819  wal 1329   wceq 1331   wcel 1480  c0 3363  EXMIDwem 4118 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 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119 This theorem is referenced by: (None)
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