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Theorem ordtri2or2exmid 4555
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 4510 . . . . 5 {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
3 suc0 4396 . . . . . 6 suc ∅ = {∅}
4 0elon 4377 . . . . . . 7 ∅ ∈ On
54onsuci 4500 . . . . . 6 suc ∅ ∈ On
63, 5eqeltrri 2244 . . . . 5 {∅} ∈ On
7 sseq1 3170 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
8 sseq2 3171 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦𝑥𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
97, 8orbi12d 788 . . . . . 6 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
10 sseq2 3171 . . . . . . 7 (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
11 sseq1 3170 . . . . . . 7 (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
1210, 11orbi12d 788 . . . . . 6 (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
139, 12rspc2va 2848 . . . . 5 ((({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
142, 6, 13mpanl12 434 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
151, 14ax-mp 5 . . 3 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
16 elirr 4525 . . . . 5 ¬ {∅} ∈ {∅}
17 simpl 108 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
18 simpr 109 . . . . . . . 8 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
19 p0ex 4174 . . . . . . . . . 10 {∅} ∈ V
2019prid2 3690 . . . . . . . . 9 {∅} ∈ {∅, {∅}}
21 biidd 171 . . . . . . . . . 10 (𝑧 = {∅} → (𝜑𝜑))
2221elrab3 2887 . . . . . . . . 9 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
2320, 22ax-mp 5 . . . . . . . 8 ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
2418, 23sylibr 133 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
2517, 24sseldd 3148 . . . . . 6 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
2625ex 114 . . . . 5 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
2716, 26mtoi 659 . . . 4 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
28 snssg 3716 . . . . . 6 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
294, 28ax-mp 5 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
30 0ex 4116 . . . . . . . 8 ∅ ∈ V
3130prid1 3689 . . . . . . 7 ∅ ∈ {∅, {∅}}
32 biidd 171 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
3332elrab3 2887 . . . . . . 7 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
3431, 33ax-mp 5 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
3534biimpi 119 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3629, 35sylbir 134 . . . 4 ({∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3727, 36orim12i 754 . . 3 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) → (¬ 𝜑𝜑))
3815, 37ax-mp 5 . 2 𝜑𝜑)
39 orcom 723 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4038, 39mpbi 144 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 703   = wceq 1348  wcel 2141  wral 2448  {crab 2452  wss 3121  c0 3414  {csn 3583  {cpr 3584  Oncon0 4348  suc csuc 4350
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by:  onintexmid  4557
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