Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmidundif Unicode version

Theorem exmidundif 4129
 Description: Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3443 and undifdcss 6811 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmidundif EXMID
Distinct variable group:   ,

Proof of Theorem exmidundif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 undifss 3443 . . . . . . . 8
21biimpi 119 . . . . . . 7
32adantl 275 . . . . . 6 EXMID
4 elun1 3243 . . . . . . . . . . 11
54adantl 275 . . . . . . . . . 10 EXMID
6 simplr 519 . . . . . . . . . . . 12 EXMID
7 simpr 109 . . . . . . . . . . . 12 EXMID
86, 7eldifd 3081 . . . . . . . . . . 11 EXMID
9 elun2 3244 . . . . . . . . . . 11
108, 9syl 14 . . . . . . . . . 10 EXMID
11 exmidexmid 4120 . . . . . . . . . . . 12 EXMID DECID
12 exmiddc 821 . . . . . . . . . . . 12 DECID
1311, 12syl 14 . . . . . . . . . . 11 EXMID
1413adantr 274 . . . . . . . . . 10 EXMID
155, 10, 14mpjaodan 787 . . . . . . . . 9 EXMID
1615ex 114 . . . . . . . 8 EXMID
1716ssrdv 3103 . . . . . . 7 EXMID
1817adantr 274 . . . . . 6 EXMID
193, 18eqssd 3114 . . . . 5 EXMID
2019ex 114 . . . 4 EXMID
21 ssun1 3239 . . . . 5
22 sseq2 3121 . . . . 5
2321, 22mpbii 147 . . . 4
2420, 23impbid1 141 . . 3 EXMID
2524alrimivv 1847 . 2 EXMID
26 vex 2689 . . . . . 6
27 p0ex 4112 . . . . . 6
28 sseq12 3122 . . . . . . . 8
29 simpl 108 . . . . . . . . . 10
30 simpr 109 . . . . . . . . . . 11
3130, 29difeq12d 3195 . . . . . . . . . 10
3229, 31uneq12d 3231 . . . . . . . . 9
3332, 30eqeq12d 2154 . . . . . . . 8
3428, 33bibi12d 234 . . . . . . 7
3534spc2gv 2776 . . . . . 6
3626, 27, 35mp2an 422 . . . . 5
37 0ex 4055 . . . . . . . 8
3837snid 3556 . . . . . . 7
39 eleq2 2203 . . . . . . 7
4038, 39mpbiri 167 . . . . . 6
41 eldifn 3199 . . . . . . . 8
4241orim2i 750 . . . . . . 7
43 elun 3217 . . . . . . 7
44 df-dc 820 . . . . . . 7 DECID
4542, 43, 443imtr4i 200 . . . . . 6 DECID
4640, 45syl 14 . . . . 5 DECID
4736, 46syl6bi 162 . . . 4 DECID
4847alrimiv 1846 . . 3 DECID
49 df-exmid 4119 . . 3 EXMID DECID
5048, 49sylibr 133 . 2 EXMID
5125, 50impbii 125 1 EXMID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wo 697  DECID wdc 819  wal 1329   wceq 1331   wcel 1480  cvv 2686   cdif 3068   cun 3069   wss 3071  c0 3363  csn 3527  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-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator