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Theorem ordsoexmid 4576
Description: Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)
Hypothesis
Ref Expression
ordsoexmid.1 E Or On
Assertion
Ref Expression
ordsoexmid (𝜑 ∨ ¬ 𝜑)

Proof of Theorem ordsoexmid
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtriexmidlem 4533 . . . . 5 {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
21elexi 2764 . . . 4 {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
32sucid 4432 . . 3 {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}
41onsuci 4530 . . . 4 suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
5 suc0 4426 . . . . 5 suc ∅ = {∅}
6 0elon 4407 . . . . . 6 ∅ ∈ On
76onsuci 4530 . . . . 5 suc ∅ ∈ On
85, 7eqeltrri 2263 . . . 4 {∅} ∈ On
9 eleq1 2252 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ On ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
1093anbi1d 1327 . . . . . 6 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
11 eleq1 2252 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
12 eleq1 2252 . . . . . . . 8 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅}))
1312orbi1d 792 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
1411, 13imbi12d 234 . . . . . 6 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))))
1510, 14imbi12d 234 . . . . 5 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))) ↔ (({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))))
164elexi 2764 . . . . . 6 suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
17 eleq1 2252 . . . . . . . 8 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ On ↔ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
18173anbi2d 1328 . . . . . . 7 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) ↔ (𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
19 eleq2 2253 . . . . . . . 8 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥𝑦𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
20 eleq2 2253 . . . . . . . . 9 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ 𝑦 ↔ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
2120orbi2d 791 . . . . . . . 8 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
2219, 21imbi12d 234 . . . . . . 7 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)) ↔ (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))))
2318, 22imbi12d 234 . . . . . 6 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦))) ↔ ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))))
24 p0ex 4203 . . . . . . 7 {∅} ∈ V
25 eleq1 2252 . . . . . . . . 9 (𝑧 = {∅} → (𝑧 ∈ On ↔ {∅} ∈ On))
26253anbi3d 1329 . . . . . . . 8 (𝑧 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On)))
27 eleq2 2253 . . . . . . . . . 10 (𝑧 = {∅} → (𝑥𝑧𝑥 ∈ {∅}))
28 eleq1 2252 . . . . . . . . . 10 (𝑧 = {∅} → (𝑧𝑦 ↔ {∅} ∈ 𝑦))
2927, 28orbi12d 794 . . . . . . . . 9 (𝑧 = {∅} → ((𝑥𝑧𝑧𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
3029imbi2d 230 . . . . . . . 8 (𝑧 = {∅} → ((𝑥𝑦 → (𝑥𝑧𝑧𝑦)) ↔ (𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦))))
3126, 30imbi12d 234 . . . . . . 7 (𝑧 = {∅} → (((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑥𝑦 → (𝑥𝑧𝑧𝑦))) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))))
32 ordsoexmid.1 . . . . . . . . . . 11 E Or On
33 df-iso 4312 . . . . . . . . . . 11 ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦))))
3432, 33mpbi 145 . . . . . . . . . 10 ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦)))
3534simpri 113 . . . . . . . . 9 𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦))
36 epel 4307 . . . . . . . . . . . 12 (𝑥 E 𝑦𝑥𝑦)
37 epel 4307 . . . . . . . . . . . . 13 (𝑥 E 𝑧𝑥𝑧)
38 epel 4307 . . . . . . . . . . . . 13 (𝑧 E 𝑦𝑧𝑦)
3937, 38orbi12i 765 . . . . . . . . . . . 12 ((𝑥 E 𝑧𝑧 E 𝑦) ↔ (𝑥𝑧𝑧𝑦))
4036, 39imbi12i 239 . . . . . . . . . . 11 ((𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦)) ↔ (𝑥𝑦 → (𝑥𝑧𝑧𝑦)))
41402ralbii 2498 . . . . . . . . . 10 (∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦)) ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥𝑦 → (𝑥𝑧𝑧𝑦)))
4241ralbii 2496 . . . . . . . . 9 (∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦)) ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥𝑦 → (𝑥𝑧𝑧𝑦)))
4335, 42mpbi 145 . . . . . . . 8 𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥𝑦 → (𝑥𝑧𝑧𝑦))
4443rspec3 2580 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑥𝑦 → (𝑥𝑧𝑧𝑦)))
4524, 31, 44vtocl 2806 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
4616, 23, 45vtocl 2806 . . . . 5 ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
472, 15, 46vtocl 2806 . . . 4 (({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
481, 4, 8, 47mp3an 1348 . . 3 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
492elsn 3623 . . . . 5 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} = ∅)
50 ordtriexmidlem2 4534 . . . . 5 ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
5149, 50sylbi 121 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
52 elirr 4555 . . . . . . 7 ¬ {∅} ∈ {∅}
53 elrabi 2905 . . . . . . 7 ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} ∈ {∅})
5452, 53mto 663 . . . . . 6 ¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑}
55 elsuci 4418 . . . . . . 7 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
5655ord 725 . . . . . 6 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
5754, 56mpi 15 . . . . 5 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑})
58 0ex 4145 . . . . . . 7 ∅ ∈ V
59 biidd 172 . . . . . . 7 (𝑤 = ∅ → (𝜑𝜑))
6058, 59rabsnt 3682 . . . . . 6 ({𝑤 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
6160eqcoms 2192 . . . . 5 ({∅} = {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
6257, 61syl 14 . . . 4 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
6351, 62orim12i 760 . . 3 (({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
643, 48, 63mp2b 8 . 2 𝜑𝜑)
65 orcom 729 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
6664, 65mpbi 145 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  w3a 980   = wceq 1364  wcel 2160  wral 2468  {crab 2472  c0 3437  {csn 3607   class class class wbr 4018   E cep 4302   Po wpo 4309   Or wor 4310  Oncon0 4378  suc csuc 4380
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-tr 4117  df-eprel 4304  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386
This theorem is referenced by: (None)
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