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Theorem ontr2exmid 4536
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 3252 . . . . 5 {𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2 p0ex 4200 . . . . . 6 {∅} ∈ V
32prid2 3711 . . . . 5 {∅} ∈ {∅, {∅}}
4 2ordpr 4535 . . . . . . 7 Ord {∅, {∅}}
5 pp0ex 4201 . . . . . . . 8 {∅, {∅}} ∈ V
65elon 4386 . . . . . . 7 ({∅, {∅}} ∈ On ↔ Ord {∅, {∅}})
74, 6mpbir 146 . . . . . 6 {∅, {∅}} ∈ On
8 ordtriexmidlem 4530 . . . . . . . 8 {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
9 ontr2exmid.1 . . . . . . . 8 𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
10 sseq1 3190 . . . . . . . . . . . . 13 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥𝑦 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦))
1110anbi1d 465 . . . . . . . . . . . 12 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦𝑦𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧)))
12 eleq1 2250 . . . . . . . . . . . 12 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥𝑧 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧))
1311, 12imbi12d 234 . . . . . . . . . . 11 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
1413ralbidv 2487 . . . . . . . . . 10 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (∀𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
1514albidv 1834 . . . . . . . . 9 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑦𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
1615rspcv 2849 . . . . . . . 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 3191 . . . . . . . . . . 11 (𝑦 = {∅} → ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
19 eleq1 2250 . . . . . . . . . . 11 (𝑦 = {∅} → (𝑦𝑧 ↔ {∅} ∈ 𝑧))
2018, 19anbi12d 473 . . . . . . . . . 10 (𝑦 = {∅} → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧)))
2120imbi1d 231 . . . . . . . . 9 (𝑦 = {∅} → ((({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
2221ralbidv 2487 . . . . . . . 8 (𝑦 = {∅} → (∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)))
232, 22spcv 2843 . . . . . . 7 (∀𝑦𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ 𝑦𝑦𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) → ∀𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧))
2417, 23ax-mp 5 . . . . . 6 𝑧 ∈ On (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧)
25 eleq2 2251 . . . . . . . . 9 (𝑧 = {∅, {∅}} → ({∅} ∈ 𝑧 ↔ {∅} ∈ {∅, {∅}}))
2625anbi2d 464 . . . . . . . 8 (𝑧 = {∅, {∅}} → (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}})))
27 eleq2 2251 . . . . . . . 8 (𝑧 = {∅, {∅}} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧 ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}}))
2826, 27imbi12d 234 . . . . . . 7 (𝑧 = {∅, {∅}} → ((({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ 𝑧) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ 𝑧) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ⊆ {∅} ∧ {∅} ∈ {∅, {∅}}) → {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}})))
2928rspcv 2849 . . . . . 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 426 . . . 4 {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}}
32 elpri 3627 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅, {∅}} → ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅}))
3331, 32ax-mp 5 . . 3 ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅})
34 ordtriexmidlem2 4531 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
35 0ex 4142 . . . . 5 ∅ ∈ V
36 biidd 172 . . . . 5 (𝑤 = ∅ → (𝜑𝜑))
3735, 36rabsnt 3679 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
3834, 37orim12i 760 . . 3 (({𝑤 ∈ {∅} ∣ 𝜑} = ∅ ∨ {𝑤 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑𝜑))
3933, 38ax-mp 5 . 2 𝜑𝜑)
40 orcom 729 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4139, 40mpbi 145 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  wal 1361   = wceq 1363  wcel 2158  wral 2465  {crab 2469  wss 3141  c0 3434  {csn 3604  {cpr 3605  Ord word 4374  Oncon0 4375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by: (None)
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