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Theorem ordtri2or2exmid 4349
Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
Hypothesis
Ref Expression
ordtri2or2exmid.1 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)
Assertion
Ref Expression
ordtri2or2exmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ordtri2or2exmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ordtri2or2exmid.1 . . . 4 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)
2 ordtri2or2exmidlem 4304 . . . . 5 {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
3 suc0 4201 . . . . . 6 suc ∅ = {∅}
4 0elon 4182 . . . . . . 7 ∅ ∈ On
54onsuci 4295 . . . . . 6 suc ∅ ∈ On
63, 5eqeltrri 2156 . . . . 5 {∅} ∈ On
7 sseq1 3031 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
8 sseq2 3032 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦𝑥𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
97, 8orbi12d 740 . . . . . 6 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
10 sseq2 3032 . . . . . . 7 (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
11 sseq1 3031 . . . . . . 7 (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
1210, 11orbi12d 740 . . . . . 6 (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
139, 12rspc2va 2724 . . . . 5 ((({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
142, 6, 13mpanl12 427 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
151, 14ax-mp 7 . . 3 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
16 elirr 4319 . . . . 5 ¬ {∅} ∈ {∅}
17 simpl 107 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
18 simpr 108 . . . . . . . 8 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
19 p0ex 3986 . . . . . . . . . 10 {∅} ∈ V
2019prid2 3523 . . . . . . . . 9 {∅} ∈ {∅, {∅}}
21 biidd 170 . . . . . . . . . 10 (𝑧 = {∅} → (𝜑𝜑))
2221elrab3 2760 . . . . . . . . 9 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
2320, 22ax-mp 7 . . . . . . . 8 ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
2418, 23sylibr 132 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
2517, 24sseldd 3011 . . . . . 6 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
2625ex 113 . . . . 5 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
2716, 26mtoi 623 . . . 4 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
28 snssg 3547 . . . . . 6 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
294, 28ax-mp 7 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
30 0ex 3931 . . . . . . . 8 ∅ ∈ V
3130prid1 3522 . . . . . . 7 ∅ ∈ {∅, {∅}}
32 biidd 170 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
3332elrab3 2760 . . . . . . 7 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
3431, 33ax-mp 7 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
3534biimpi 118 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3629, 35sylbir 133 . . . 4 ({∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3727, 36orim12i 709 . . 3 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) → (¬ 𝜑𝜑))
3815, 37ax-mp 7 . 2 𝜑𝜑)
39 orcom 680 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4038, 39mpbi 143 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  wral 2353  {crab 2357  wss 2984  c0 3269  {csn 3422  {cpr 3423  Oncon0 4153  suc csuc 4155
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-13 1445  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  ax-pr 3999  ax-un 4223  ax-setind 4315
This theorem depends on definitions:  df-bi 115  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-ne 2250  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-pr 3429  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158  df-suc 4161
This theorem is referenced by:  onintexmid  4350
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