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Theorem ontri2orexmidim 4554
Description: Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4553. (Contributed by Jim Kingdon, 26-Aug-2024.)
Assertion
Ref Expression
ontri2orexmidim (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → DECID 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ontri2orexmidim
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ordtri2or2exmidlem 4508 . . . . 5 {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
2 suc0 4394 . . . . . 6 suc ∅ = {∅}
3 0elon 4375 . . . . . . 7 ∅ ∈ On
43onsuci 4498 . . . . . 6 suc ∅ ∈ On
52, 4eqeltrri 2244 . . . . 5 {∅} ∈ On
6 sseq1 3170 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
7 sseq2 3171 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦𝑥𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
86, 7orbi12d 788 . . . . . 6 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
9 sseq2 3171 . . . . . . 7 (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
10 sseq1 3170 . . . . . . 7 (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
119, 10orbi12d 788 . . . . . 6 (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
128, 11rspc2va 2848 . . . . 5 ((({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
131, 5, 12mpanl12 434 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
145onirri 4525 . . . . . 6 ¬ {∅} ∈ {∅}
15 simpl 108 . . . . . . . 8 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
16 simpr 109 . . . . . . . . 9 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
17 p0ex 4172 . . . . . . . . . . 11 {∅} ∈ V
1817prid2 3688 . . . . . . . . . 10 {∅} ∈ {∅, {∅}}
19 biidd 171 . . . . . . . . . . 11 (𝑧 = {∅} → (𝜑𝜑))
2019elrab3 2887 . . . . . . . . . 10 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
2118, 20ax-mp 5 . . . . . . . . 9 ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
2216, 21sylibr 133 . . . . . . . 8 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
2315, 22sseldd 3148 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
2423ex 114 . . . . . 6 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
2514, 24mtoi 659 . . . . 5 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
26 snssg 3714 . . . . . . 7 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
273, 26ax-mp 5 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
28 0ex 4114 . . . . . . . 8 ∅ ∈ V
2928prid1 3687 . . . . . . 7 ∅ ∈ {∅, {∅}}
30 biidd 171 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
3130elrab3 2887 . . . . . . 7 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
3229, 31ax-mp 5 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
3327, 32sylbb1 136 . . . . 5 ({∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3425, 33orim12i 754 . . . 4 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) → (¬ 𝜑𝜑))
3513, 34syl 14 . . 3 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → (¬ 𝜑𝜑))
3635orcomd 724 . 2 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → (𝜑 ∨ ¬ 𝜑))
37 df-dc 830 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
3836, 37sylibr 133 1 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 703  DECID wdc 829   = wceq 1348  wcel 2141  wral 2448  {crab 2452  wss 3121  c0 3414  {csn 3581  {cpr 3582  Oncon0 4346  suc csuc 4348
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 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-dc 830  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 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351  df-suc 4354
This theorem is referenced by:  exmidontri2or  7207
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