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Theorem ordtriexmid 4445
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 3372 . . . 4 ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2 ordtriexmidlem 4443 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 eleq1 2203 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
4 eqeq1 2147 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
5 eleq2 2204 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
63, 4, 53orbi123d 1290 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
7 0elon 4322 . . . . . . . 8 ∅ ∈ On
8 0ex 4063 . . . . . . . . 9 ∅ ∈ V
9 eleq1 2203 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦 ∈ On ↔ ∅ ∈ On))
109anbi2d 460 . . . . . . . . . 10 (𝑦 = ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ ∅ ∈ On)))
11 eleq2 2204 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑥𝑦𝑥 ∈ ∅))
12 eqeq2 2150 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑥 = 𝑦𝑥 = ∅))
13 eleq1 2203 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦𝑥 ↔ ∅ ∈ 𝑥))
1411, 12, 133orbi123d 1290 . . . . . . . . . 10 (𝑦 = ∅ → ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥)))
1510, 14imbi12d 233 . . . . . . . . 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 2524 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
188, 15, 17vtocl 2743 . . . . . . . 8 ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
197, 18mpan2 422 . . . . . . 7 (𝑥 ∈ On → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
206, 19vtoclga 2755 . . . . . 6 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
212, 20ax-mp 5 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
22 3orass 966 . . . . 5 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
2321, 22mpbi 144 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
241, 23mtpor 1404 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
25 ordtriexmidlem2 4444 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
268snid 3563 . . . . . 6 ∅ ∈ {∅}
27 biidd 171 . . . . . . 7 (𝑧 = ∅ → (𝜑𝜑))
2827elrab3 2845 . . . . . 6 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2926, 28ax-mp 5 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
3029biimpi 119 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
3125, 30orim12i 749 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
3224, 31ax-mp 5 . 2 𝜑𝜑)
33 orcom 718 . 2 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
3432, 33mpbir 145 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  w3o 962   = wceq 1332  wcel 1481  wral 2417  {crab 2421  c0 3368  {csn 3532  Oncon0 4293
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301
This theorem is referenced by: (None)
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