ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmid0el GIF version

Theorem exmid0el 4167
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 4159 . . 3 (EXMIDDECID ∅ ∈ 𝑥)
21alrimiv 1854 . 2 (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3 ax-1 6 . . . 4 (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
43alimi 1435 . . 3 (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5 df-exmid 4158 . . 3 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
64, 5sylibr 133 . 2 (∀𝑥DECID ∅ ∈ 𝑥EXMID)
72, 6impbii 125 1 (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  DECID wdc 820  wal 1333  wcel 2128  wss 3102  c0 3395  {csn 3561  EXMIDwem 4157
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-nul 4092  ax-pow 4137
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-exmid 4158
This theorem is referenced by:  exmidel  4168
  Copyright terms: Public domain W3C validator