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Theorem exmid0el 4033
Description: Excluded middle is equivalent to decidability of being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmid0el (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)

Proof of Theorem exmid0el
StepHypRef Expression
1 exmidexmid 4031 . . 3 (EXMIDDECID ∅ ∈ 𝑥)
21alrimiv 1802 . 2 (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3 ax-1 5 . . . 4 (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
43alimi 1389 . . 3 (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5 df-exmid 4030 . . 3 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
64, 5sylibr 132 . 2 (∀𝑥DECID ∅ ∈ 𝑥EXMID)
72, 6impbii 124 1 (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  DECID wdc 780  wal 1287  wcel 1438  wss 2999  c0 3286  {csn 3446  EXMIDwem 4029
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 3957  ax-nul 3965  ax-pow 4009
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 3001  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-exmid 4030
This theorem is referenced by:  exmidel  4034
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