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Theorem ordtri2orexmid 4505
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 4501 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 suc0 4394 . . . . . 6 suc ∅ = {∅}
4 0elon 4375 . . . . . . 7 ∅ ∈ On
54onsuci 4498 . . . . . 6 suc ∅ ∈ On
63, 5eqeltrri 2244 . . . . 5 {∅} ∈ On
7 eleq1 2233 . . . . . . 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 eleq2 2234 . . . . . . 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 elsni 3599 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
17 ordtriexmidlem2 4502 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1816, 17syl 14 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
19 snssg 3714 . . . . . 6 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
204, 19ax-mp 5 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
21 0ex 4114 . . . . . . . 8 ∅ ∈ V
2221snid 3612 . . . . . . 7 ∅ ∈ {∅}
23 biidd 171 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
2423elrab3 2887 . . . . . . 7 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2522, 24ax-mp 5 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2625biimpi 119 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2720, 26sylbir 134 . . . 4 ({∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2818, 27orim12i 754 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
2915, 28ax-mp 5 . 2 𝜑𝜑)
30 orcom 723 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
3129, 30mpbi 144 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wo 703   = wceq 1348  wcel 2141  wral 2448  {crab 2452  wss 3121  c0 3414  {csn 3581  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
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-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: (None)
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