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Theorem ordtri2or2exmid 4582
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 4537 . . . . 5 {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
3 suc0 4423 . . . . . 6 suc ∅ = {∅}
4 0elon 4404 . . . . . . 7 ∅ ∈ On
54onsuci 4527 . . . . . 6 suc ∅ ∈ On
63, 5eqeltrri 2261 . . . . 5 {∅} ∈ On
7 sseq1 3190 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
8 sseq2 3191 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦𝑥𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
97, 8orbi12d 794 . . . . . 6 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
10 sseq2 3191 . . . . . . 7 (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
11 sseq1 3190 . . . . . . 7 (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
1210, 11orbi12d 794 . . . . . 6 (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
139, 12rspc2va 2867 . . . . 5 ((({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
142, 6, 13mpanl12 436 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
151, 14ax-mp 5 . . 3 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
16 elirr 4552 . . . . 5 ¬ {∅} ∈ {∅}
17 simpl 109 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
18 simpr 110 . . . . . . . 8 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
19 p0ex 4200 . . . . . . . . . 10 {∅} ∈ V
2019prid2 3711 . . . . . . . . 9 {∅} ∈ {∅, {∅}}
21 biidd 172 . . . . . . . . . 10 (𝑧 = {∅} → (𝜑𝜑))
2221elrab3 2906 . . . . . . . . 9 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
2320, 22ax-mp 5 . . . . . . . 8 ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
2418, 23sylibr 134 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
2517, 24sseldd 3168 . . . . . 6 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
2625ex 115 . . . . 5 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
2716, 26mtoi 665 . . . 4 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
28 snssg 3738 . . . . . 6 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
294, 28ax-mp 5 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
30 0ex 4142 . . . . . . . 8 ∅ ∈ V
3130prid1 3710 . . . . . . 7 ∅ ∈ {∅, {∅}}
32 biidd 172 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
3332elrab3 2906 . . . . . . 7 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
3431, 33ax-mp 5 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
3534biimpi 120 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3629, 35sylbir 135 . . . 4 ({∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3727, 36orim12i 760 . . 3 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) → (¬ 𝜑𝜑))
3815, 37ax-mp 5 . 2 𝜑𝜑)
39 orcom 729 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4038, 39mpbi 145 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wo 709   = wceq 1363  wcel 2158  wral 2465  {crab 2469  wss 3141  c0 3434  {csn 3604  {cpr 3605  Oncon0 4375  suc csuc 4377
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by:  onintexmid  4584
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