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Theorem exmidel 4205
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 4196 . . 3 (EXMIDDECID 𝑥𝑦)
21alrimivv 1875 . 2 (EXMID → ∀𝑥𝑦DECID 𝑥𝑦)
3 0ex 4130 . . . 4 ∅ ∈ V
4 eleq1 2240 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
54dcbid 838 . . . . 5 (𝑥 = ∅ → (DECID 𝑥𝑦DECID ∅ ∈ 𝑦))
65albidv 1824 . . . 4 (𝑥 = ∅ → (∀𝑦DECID 𝑥𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦))
73, 6spcv 2831 . . 3 (∀𝑥𝑦DECID 𝑥𝑦 → ∀𝑦DECID ∅ ∈ 𝑦)
8 exmid0el 4204 . . 3 (EXMID ↔ ∀𝑦DECID ∅ ∈ 𝑦)
97, 8sylibr 134 . 2 (∀𝑥𝑦DECID 𝑥𝑦EXMID)
102, 9impbii 126 1 (EXMID ↔ ∀𝑥𝑦DECID 𝑥𝑦)
Colors of variables: wff set class
Syntax hints:  wb 105  DECID wdc 834  wal 1351   = wceq 1353  wcel 2148  c0 3422  EXMIDwem 4194
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-exmid 4195
This theorem is referenced by: (None)
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