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Theorem ordtri2orexmid 4367
Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
Hypothesis
Ref Expression
ordtri2orexmid.1 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)
Assertion
Ref Expression
ordtri2orexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ordtri2orexmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ordtri2orexmid.1 . . . 4 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)
2 ordtriexmidlem 4364 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 suc0 4262 . . . . . 6 suc ∅ = {∅}
4 0elon 4243 . . . . . . 7 ∅ ∈ On
54onsuci 4361 . . . . . 6 suc ∅ ∈ On
63, 5eqeltrri 2168 . . . . 5 {∅} ∈ On
7 eleq1 2157 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
8 sseq2 3063 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦𝑥𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
97, 8orbi12d 745 . . . . . 6 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
10 eleq2 2158 . . . . . . 7 (𝑦 = {∅} → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅}))
11 sseq1 3062 . . . . . . 7 (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
1210, 11orbi12d 745 . . . . . 6 (𝑦 = {∅} → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
139, 12rspc2va 2749 . . . . 5 ((({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
142, 6, 13mpanl12 428 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
151, 14ax-mp 7 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
16 elsni 3484 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
17 ordtriexmidlem2 4365 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1816, 17syl 14 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
19 snssg 3595 . . . . . 6 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
204, 19ax-mp 7 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
21 0ex 3987 . . . . . . . 8 ∅ ∈ V
2221snid 3495 . . . . . . 7 ∅ ∈ {∅}
23 biidd 171 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
2423elrab3 2786 . . . . . . 7 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2522, 24ax-mp 7 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2625biimpi 119 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2720, 26sylbir 134 . . . 4 ({∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2818, 27orim12i 714 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
2915, 28ax-mp 7 . 2 𝜑𝜑)
30 orcom 685 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
3129, 30mpbi 144 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wo 667   = wceq 1296  wcel 1445  wral 2370  {crab 2374  wss 3013  c0 3302  {csn 3466  Oncon0 4214  suc csuc 4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222
This theorem is referenced by: (None)
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