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Theorem ordtri2orexmid 4248
 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 4245 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 suc0 4148 . . . . . 6 suc ∅ = {∅}
4 0elon 4129 . . . . . . 7 ∅ ∈ On
54onsuci 4242 . . . . . 6 suc ∅ ∈ On
63, 5eqeltrri 2111 . . . . 5 {∅} ∈ On
7 eleq1 2100 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦))
8 sseq2 2967 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦𝑥𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
97, 8orbi12d 707 . . . . . 6 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
10 eleq2 2101 . . . . . . 7 (𝑦 = {∅} → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅}))
11 sseq1 2966 . . . . . . 7 (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
1210, 11orbi12d 707 . . . . . 6 (𝑦 = {∅} → (({𝑧 ∈ {∅} ∣ 𝜑} ∈ 𝑦𝑦 ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})))
139, 12rspc2va 2663 . . . . 5 ((({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
142, 6, 13mpanl12 412 . . . 4 (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
151, 14ax-mp 7 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
16 elsni 3393 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
17 ordtriexmidlem2 4246 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1816, 17syl 14 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
19 snssg 3500 . . . . . 6 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}))
204, 19ax-mp 7 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑})
21 0ex 3884 . . . . . . . 8 ∅ ∈ V
2221snid 3402 . . . . . . 7 ∅ ∈ {∅}
23 biidd 161 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
2423elrab3 2699 . . . . . . 7 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2522, 24ax-mp 7 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2625biimpi 113 . . . . 5 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2720, 26sylbir 125 . . . 4 ({∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
2818, 27orim12i 676 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
2915, 28ax-mp 7 . 2 𝜑𝜑)
30 orcom 647 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
3129, 30mpbi 133 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310   ⊆ wss 2917  ∅c0 3224  {csn 3375  Oncon0 4100  suc csuc 4102 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by: (None)
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