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

Theorem exmidel 4161
 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 ↔ ∀𝑥𝑦DECID 𝑥𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidel
StepHypRef Expression
1 exmidexmid 4152 . . 3 (EXMIDDECID 𝑥𝑦)
21alrimivv 1852 . 2 (EXMID → ∀𝑥𝑦DECID 𝑥𝑦)
3 0ex 4087 . . . 4 ∅ ∈ V
4 eleq1 2217 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
54dcbid 824 . . . . 5 (𝑥 = ∅ → (DECID 𝑥𝑦DECID ∅ ∈ 𝑦))
65albidv 1801 . . . 4 (𝑥 = ∅ → (∀𝑦DECID 𝑥𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦))
73, 6spcv 2803 . . 3 (∀𝑥𝑦DECID 𝑥𝑦 → ∀𝑦DECID ∅ ∈ 𝑦)
8 exmid0el 4160 . . 3 (EXMID ↔ ∀𝑦DECID ∅ ∈ 𝑦)
97, 8sylibr 133 . 2 (∀𝑥𝑦DECID 𝑥𝑦EXMID)
102, 9impbii 125 1 (EXMID ↔ ∀𝑥𝑦DECID 𝑥𝑦)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  DECID wdc 820  ∀wal 1330   = wceq 1332   ∈ wcel 2125  ∅c0 3390  EXMIDwem 4150 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rab 2441  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-exmid 4151 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator