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Theorem ontr2exmid 4307
Description: An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
Hypothesis
Ref Expression
ontr2exmid.1 𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
Assertion
Ref Expression
ontr2exmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦,𝑧

Proof of Theorem ontr2exmid
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3092 . . . . 5 {𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2 p0ex 3990 . . . . . 6 {∅} ∈ V
32prid2 3526 . . . . 5 {∅} ∈ {∅, {∅}}
4 2ordpr 4306 . . . . . . 7 Ord {∅, {∅}}
5 pp0ex 3991 . . . . . . . 8 {∅, {∅}} ∈ V
65elon 4168 . . . . . . 7 ({∅, {∅}} ∈ On ↔ Ord {∅, {∅}})
74, 6mpbir 144 . . . . . 6 {∅, {∅}} ∈ On
8 ordtriexmidlem 4302 . . . . . . . 8 {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
9 ontr2exmid.1 . . . . . . . 8 𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
10 sseq1 3033 . . . . . . . . . . . . 13 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦))
1110anbi1d 453 . . . . . . . . . . . 12 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦𝑦𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧)))
12 eleq1 2147 . . . . . . . . . . . 12 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥𝑧 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧))
1311, 12imbi12d 232 . . . . . . . . . . 11 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
1413ralbidv 2376 . . . . . . . . . 10 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (∀𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
1514albidv 1749 . . . . . . . . 9 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑦𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
1615rspcv 2710 . . . . . . . 8 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑦𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
178, 9, 16mp2 16 . . . . . . 7 𝑦𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)
18 sseq2 3034 . . . . . . . . . . 11 (𝑦 = {∅} → ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
19 eleq1 2147 . . . . . . . . . . 11 (𝑦 = {∅} → (𝑦𝑧 ↔ {∅} ∈ 𝑧))
2018, 19anbi12d 457 . . . . . . . . . 10 (𝑦 = {∅} → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧)))
2120imbi1d 229 . . . . . . . . 9 (𝑦 = {∅} → ((({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
2221ralbidv 2376 . . . . . . . 8 (𝑦 = {∅} → (∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
232, 22spcv 2704 . . . . . . 7 (∀𝑦𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) → ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧))
2417, 23ax-mp 7 . . . . . 6 𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)
25 eleq2 2148 . . . . . . . . 9 (𝑧 = {∅, {∅}} → ({∅} ∈ 𝑧 ↔ {∅} ∈ {∅, {∅}}))
2625anbi2d 452 . . . . . . . 8 (𝑧 = {∅, {∅}} → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}})))
27 eleq2 2148 . . . . . . . 8 (𝑧 = {∅, {∅}} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}}))
2826, 27imbi12d 232 . . . . . . 7 (𝑧 = {∅, {∅}} → ((({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}}) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}})))
2928rspcv 2710 . . . . . 6 ({∅, {∅}} ∈ On → (∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}}) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}})))
307, 24, 29mp2 16 . . . . 5 (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}}) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}})
311, 3, 30mp2an 417 . . . 4 {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}}
32 elpri 3448 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}} → ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅}))
3331, 32ax-mp 7 . . 3 ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅})
34 ordtriexmidlem2 4303 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
35 0ex 3934 . . . . 5 ∅ ∈ V
36 biidd 170 . . . . 5 (𝑤 = ∅ → (𝜑𝜑))
3735, 36rabsnt 3494 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
3834, 37orim12i 709 . . 3 (({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑𝜑))
3933, 38ax-mp 7 . 2 𝜑𝜑)
40 orcom 680 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4139, 40mpbi 143 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  wal 1285   = wceq 1287  wcel 1436  wral 2355  {crab 2359  wss 2986  c0 3272  {csn 3425  {cpr 3426  Ord word 4156  Oncon0 4157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162  df-suc 4165
This theorem is referenced by: (None)
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