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Theorem ordtri2or2exmid 4266
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 4223 . . . . 5 {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
3 suc0 4120 . . . . . 6 suc ∅ = {∅}
4 0elon 4101 . . . . . . 7 ∅ ∈ On
54onsuci 4214 . . . . . 6 suc ∅ ∈ On
63, 5eqeltrri 2111 . . . . 5 {∅} ∈ On
7 sseq1 2963 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦))
8 sseq2 2964 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → (𝑦𝑥𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
97, 8orbi12d 707 . . . . . 6 (𝑥 = {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → ((𝑥𝑦𝑦𝑥) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
10 sseq2 2964 . . . . . . 7 (𝑦 = {∅} → ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅}))
11 sseq1 2963 . . . . . . 7 (𝑦 = {∅} → (𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
1210, 11orbi12d 707 . . . . . 6 (𝑦 = {∅} → (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ 𝑦𝑦 ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})))
139, 12rspc2va 2660 . . . . 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 elirr 4236 . . . . 5 ¬ {∅} ∈ {∅}
17 simpl 102 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅})
18 simpr 103 . . . . . . . 8 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → 𝜑)
19 p0ex 3936 . . . . . . . . . 10 {∅} ∈ V
2019prid2 3474 . . . . . . . . 9 {∅} ∈ {∅, {∅}}
21 biidd 161 . . . . . . . . . 10 (𝑧 = {∅} → (𝜑𝜑))
2221elrab3 2696 . . . . . . . . 9 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
2320, 22ax-mp 7 . . . . . . . 8 ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
2418, 23sylibr 137 . . . . . . 7 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
2517, 24sseldd 2943 . . . . . 6 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∧ 𝜑) → {∅} ∈ {∅})
2625ex 108 . . . . 5 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → (𝜑 → {∅} ∈ {∅}))
2716, 26mtoi 590 . . . 4 ({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} → ¬ 𝜑)
28 snssg 3497 . . . . . 6 (∅ ∈ On → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}))
294, 28ax-mp 7 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑})
30 0ex 3881 . . . . . . . 8 ∅ ∈ V
3130prid1 3473 . . . . . . 7 ∅ ∈ {∅, {∅}}
32 biidd 161 . . . . . . . 8 (𝑧 = ∅ → (𝜑𝜑))
3332elrab3 2696 . . . . . . 7 (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑))
3431, 33ax-mp 7 . . . . . 6 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} ↔ 𝜑)
3534biimpi 113 . . . . 5 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3629, 35sylbir 125 . . . 4 ({∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
3727, 36orim12i 676 . . 3 (({𝑧 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅} ∨ {∅} ⊆ {𝑧 ∈ {∅, {∅}} ∣ 𝜑}) → (¬ 𝜑𝜑))
3815, 37ax-mp 7 . 2 𝜑𝜑)
39 orcom 647 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4038, 39mpbi 133 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97  wb 98  wo 629   = wceq 1243  wcel 1393  wral 2303  {crab 2307  wss 2914  c0 3221  {csn 3372  {cpr 3373  Oncon0 4072  suc csuc 4074
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 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232
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-ne 2206  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-uni 3578  df-tr 3852  df-iord 4075  df-on 4077  df-suc 4080
This theorem is referenced by:  onintexmid  4267
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