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Theorem ontriexmidim 4515
Description: Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4514. (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 3424 . . . . . 6 ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
21a1i 9 . . . . 5 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅)
3 ordtriexmidlem 4512 . . . . . . . 8 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
4 0elon 4386 . . . . . . . 8 ∅ ∈ On
5 eleq1 2238 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
6 eqeq1 2182 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦))
7 eleq2 2239 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦𝑥𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
85, 6, 73orbi123d 1311 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
9 eleq2 2239 . . . . . . . . . 10 (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
10 eqeq2 2185 . . . . . . . . . 10 (𝑦 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} = 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
11 eleq1 2238 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
129, 10, 113orbi123d 1311 . . . . . . . . 9 (𝑦 = ∅ → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ∨ {𝑧 ∈ {∅} ∣ 𝜑} = 𝑦𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
138, 12rspc2v 2852 . . . . . . . 8 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ ∅ ∈ On) → (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
143, 4, 13mp2an 426 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
15 3orass 981 . . . . . . 7 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
1614, 15sylib 122 . . . . . 6 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
1716orcomd 729 . . . . 5 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ∨ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
182, 17ecased 1349 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
19 ordtriexmidlem2 4513 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
20 0ex 4125 . . . . . . . 8 ∅ ∈ V
2120snid 3620 . . . . . . 7 ∅ ∈ {∅}
22 biidd 172 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
2322elrab3 2892 . . . . . . 7 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2421, 23ax-mp 5 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2524biimpi 120 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2619, 25orim12i 759 . . . 4 (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
2718, 26syl 14 . . 3 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → (¬ 𝜑𝜑))
2827orcomd 729 . 2 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → (𝜑 ∨ ¬ 𝜑))
29 df-dc 835 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
3028, 29sylibr 134 1 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708  DECID wdc 834  w3o 977   = wceq 1353  wcel 2146  wral 2453  {crab 2457  c0 3420  {csn 3589  Oncon0 4357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-uni 3806  df-tr 4097  df-iord 4360  df-on 4362  df-suc 4365
This theorem is referenced by:  exmidontri  7228
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