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Theorem ordtriexmid 4300
Description: Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

(Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

Hypothesis
Ref Expression
ordtriexmid.1 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥)
Assertion
Ref Expression
ordtriexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem ordtriexmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 noel 3273 . . . 4 ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2 ordtriexmidlem 4298 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 eleq1 2145 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
4 eqeq1 2089 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
5 eleq2 2146 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
63, 4, 53orbi123d 1243 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
7 0elon 4182 . . . . . . . 8 ∅ ∈ On
8 0ex 3931 . . . . . . . . 9 ∅ ∈ V
9 eleq1 2145 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦 ∈ On ↔ ∅ ∈ On))
109anbi2d 452 . . . . . . . . . 10 (𝑦 = ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ ∅ ∈ On)))
11 eleq2 2146 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑥𝑦𝑥 ∈ ∅))
12 eqeq2 2092 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑥 = 𝑦𝑥 = ∅))
13 eleq1 2145 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦𝑥 ↔ ∅ ∈ 𝑥))
1411, 12, 133orbi123d 1243 . . . . . . . . . 10 (𝑦 = ∅ → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥)))
1510, 14imbi12d 232 . . . . . . . . 9 (𝑦 = ∅ → (((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥)) ↔ ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))))
16 ordtriexmid.1 . . . . . . . . . 10 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥)
1716rspec2 2456 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
188, 15, 17vtocl 2664 . . . . . . . 8 ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
197, 18mpan2 416 . . . . . . 7 (𝑥 ∈ On → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
206, 19vtoclga 2675 . . . . . 6 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
212, 20ax-mp 7 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
22 3orass 923 . . . . 5 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
2321, 22mpbi 143 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
241, 23mtpor 1357 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
25 ordtriexmidlem2 4299 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
268snid 3449 . . . . . 6 ∅ ∈ {∅}
27 biidd 170 . . . . . . 7 (𝑧 = ∅ → (𝜑𝜑))
2827elrab3 2760 . . . . . 6 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2926, 28ax-mp 7 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
3029biimpi 118 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
3125, 30orim12i 709 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
3224, 31ax-mp 7 . 2 𝜑𝜑)
33 orcom 680 . 2 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
3432, 33mpbir 144 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3o 919   = wceq 1285  wcel 1434  wral 2353  {crab 2357  c0 3269  {csn 3422  Oncon0 4153
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158  df-suc 4161
This theorem is referenced by: (None)
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