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Theorem exmidel 4025
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 4022 . . 3 (EXMIDDECID 𝑥𝑦)
21alrimivv 1803 . 2 (EXMID → ∀𝑥𝑦DECID 𝑥𝑦)
3 0ex 3958 . . . 4 ∅ ∈ V
4 eleq1 2150 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
54dcbid 786 . . . . 5 (𝑥 = ∅ → (DECID 𝑥𝑦DECID ∅ ∈ 𝑦))
65albidv 1752 . . . 4 (𝑥 = ∅ → (∀𝑦DECID 𝑥𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦))
73, 6spcv 2712 . . 3 (∀𝑥𝑦DECID 𝑥𝑦 → ∀𝑦DECID ∅ ∈ 𝑦)
8 exmid0el 4024 . . 3 (EXMID ↔ ∀𝑦DECID ∅ ∈ 𝑦)
97, 8sylibr 132 . 2 (∀𝑥𝑦DECID 𝑥𝑦EXMID)
102, 9impbii 124 1 (EXMID ↔ ∀𝑥𝑦DECID 𝑥𝑦)
Colors of variables: wff set class
Syntax hints:  wb 103  DECID wdc 780  wal 1287   = wceq 1289  wcel 1438  c0 3284  EXMIDwem 4020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-exmid 4021
This theorem is referenced by: (None)
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