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

Theorem ontriexmidim 4481
Description: Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4480. (Contributed by Jim Kingdon, 26-Aug-2024.)
Assertion
Ref Expression
ontriexmidim (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → DECID 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ontriexmidim
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 noel 3398 . . . . . 6 ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
21a1i 9 . . . . 5 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅)
3 ordtriexmidlem 4478 . . . . . . . 8 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
4 0elon 4352 . . . . . . . 8 ∅ ∈ On
5 eleq1 2220 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
6 eqeq1 2164 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦))
7 eleq2 2221 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦𝑥𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
85, 6, 73orbi123d 1293 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
9 eleq2 2221 . . . . . . . . . 10 (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
10 eqeq2 2167 . . . . . . . . . 10 (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
11 eleq1 2220 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
129, 10, 113orbi123d 1293 . . . . . . . . 9 (𝑦 = ∅ → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
138, 12rspc2v 2829 . . . . . . . 8 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ ∅ ∈ On) → (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
143, 4, 13mp2an 423 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
15 3orass 966 . . . . . . 7 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
1614, 15sylib 121 . . . . . 6 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
1716orcomd 719 . . . . 5 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ∨ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
182, 17ecased 1331 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
19 ordtriexmidlem2 4479 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
20 0ex 4091 . . . . . . . 8 ∅ ∈ V
2120snid 3591 . . . . . . 7 ∅ ∈ {∅}
22 biidd 171 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
2322elrab3 2869 . . . . . . 7 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2421, 23ax-mp 5 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2524biimpi 119 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2619, 25orim12i 749 . . . 4 (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
2718, 26syl 14 . . 3 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → (¬ 𝜑𝜑))
2827orcomd 719 . 2 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → (𝜑 ∨ ¬ 𝜑))
29 df-dc 821 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
3028, 29sylibr 133 1 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 698  DECID wdc 820  w3o 962   = wceq 1335  wcel 2128  wral 2435  {crab 2439  c0 3394  {csn 3560  Oncon0 4323
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 4082  ax-nul 4090  ax-pow 4135
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331
This theorem is referenced by:  exmidontri  7174
  Copyright terms: Public domain W3C validator