ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pw1dc1 GIF version

Theorem pw1dc1 6891
Description: If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
Assertion
Ref Expression
pw1dc1 (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)

Proof of Theorem pw1dc1
StepHypRef Expression
1 pw1dc0el 6889 . 2 (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
2 elpwi 3575 . . . . 5 (𝑥 ∈ 𝒫 1o𝑥 ⊆ 1o)
3 ss1o0el1o 6890 . . . . 5 (𝑥 ⊆ 1o → (∅ ∈ 𝑥𝑥 = 1o))
42, 3syl 14 . . . 4 (𝑥 ∈ 𝒫 1o → (∅ ∈ 𝑥𝑥 = 1o))
54dcbid 833 . . 3 (𝑥 ∈ 𝒫 1o → (DECID ∅ ∈ 𝑥DECID 𝑥 = 1o))
65ralbiia 2484 . 2 (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
71, 6bitri 183 1 (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
Colors of variables: wff set class
Syntax hints:  wb 104  DECID wdc 829   = wceq 1348  wcel 2141  wral 2448  wss 3121  c0 3414  𝒫 cpw 3566  EXMIDwem 4180  1oc1o 6388
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-exmid 4181  df-suc 4356  df-1o 6395
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator