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

Theorem dcextest 4386
Description: If it is decidable whether {𝑥𝜑} is a set, then ¬ 𝜑 is decidable (where 𝑥 does not occur in 𝜑). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition ¬ 𝜑 is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
Hypothesis
Ref Expression
dcextest.ex DECID {𝑥𝜑} ∈ V
Assertion
Ref Expression
dcextest DECID ¬ 𝜑
Distinct variable group:   𝜑,𝑥

Proof of Theorem dcextest
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dcextest.ex . . . 4 DECID {𝑥𝜑} ∈ V
2 exmiddc 782 . . . 4 (DECID {𝑥𝜑} ∈ V → ({𝑥𝜑} ∈ V ∨ ¬ {𝑥𝜑} ∈ V))
31, 2ax-mp 7 . . 3 ({𝑥𝜑} ∈ V ∨ ¬ {𝑥𝜑} ∈ V)
4 vprc 3963 . . . . . . 7 ¬ V ∈ V
5 df-v 2621 . . . . . . . . 9 V = {𝑥𝑥 = 𝑥}
6 equid 1634 . . . . . . . . . . 11 𝑥 = 𝑥
7 pm5.1im 171 . . . . . . . . . . 11 (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥𝜑)))
86, 7ax-mp 7 . . . . . . . . . 10 (𝜑 → (𝑥 = 𝑥𝜑))
98abbidv 2205 . . . . . . . . 9 (𝜑 → {𝑥𝑥 = 𝑥} = {𝑥𝜑})
105, 9syl5req 2133 . . . . . . . 8 (𝜑 → {𝑥𝜑} = V)
1110eleq1d 2156 . . . . . . 7 (𝜑 → ({𝑥𝜑} ∈ V ↔ V ∈ V))
124, 11mtbiri 635 . . . . . 6 (𝜑 → ¬ {𝑥𝜑} ∈ V)
1312con2i 592 . . . . 5 ({𝑥𝜑} ∈ V → ¬ 𝜑)
14 vex 2622 . . . . . . . . . 10 𝑦 ∈ V
15 biidd 170 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜑))
1614, 15elab 2758 . . . . . . . . 9 (𝑦 ∈ {𝑥𝜑} ↔ 𝜑)
1716notbii 629 . . . . . . . 8 𝑦 ∈ {𝑥𝜑} ↔ ¬ 𝜑)
1817biimpri 131 . . . . . . 7 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
1918eq0rdv 3324 . . . . . 6 𝜑 → {𝑥𝜑} = ∅)
20 0ex 3958 . . . . . 6 ∅ ∈ V
2119, 20syl6eqel 2178 . . . . 5 𝜑 → {𝑥𝜑} ∈ V)
2213, 21impbii 124 . . . 4 ({𝑥𝜑} ∈ V ↔ ¬ 𝜑)
2322notbii 629 . . . 4 (¬ {𝑥𝜑} ∈ V ↔ ¬ ¬ 𝜑)
2422, 23orbi12i 716 . . 3 (({𝑥𝜑} ∈ V ∨ ¬ {𝑥𝜑} ∈ V) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
253, 24mpbi 143 . 2 𝜑 ∨ ¬ ¬ 𝜑)
26 dftest 860 . 2 (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
2725, 26mpbir 144 1 DECID ¬ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 664  DECID wdc 780  wcel 1438  {cab 2074  Vcvv 2619  c0 3284
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator