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Theorem ordsoexmid 4286
 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 4245 . . . . 5 {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
21elexi 2567 . . . 4 {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
32sucid 4154 . . 3 {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}
41onsuci 4242 . . . 4 suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On
5 suc0 4148 . . . . 5 suc ∅ = {∅}
6 0elon 4129 . . . . . 6 ∅ ∈ On
76onsuci 4242 . . . . 5 suc ∅ ∈ On
85, 7eqeltrri 2111 . . . 4 {∅} ∈ On
9 eleq1 2100 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ On ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
1093anbi1d 1211 . . . . . 6 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
11 eleq1 2100 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
12 eleq1 2100 . . . . . . . 8 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅}))
1312orbi1d 705 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
1411, 13imbi12d 223 . . . . . 6 (𝑥 = {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})) ↔ ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))))
1510, 14imbi12d 223 . . . . 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 2567 . . . . . 6 suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ V
17 eleq1 2100 . . . . . . . 8 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ On ↔ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On))
18173anbi2d 1212 . . . . . . 7 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) ↔ (𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On)))
19 eleq2 2101 . . . . . . . 8 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥𝑦𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
20 eleq2 2101 . . . . . . . . 9 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ 𝑦 ↔ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
2120orbi2d 704 . . . . . . . 8 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
2219, 21imbi12d 223 . . . . . . 7 (𝑦 = suc {𝑤 ∈ {∅} ∣ 𝜑} → ((𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)) ↔ (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))))
2318, 22imbi12d 223 . . . . . 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 3939 . . . . . . 7 {∅} ∈ V
25 eleq1 2100 . . . . . . . . 9 (𝑧 = {∅} → (𝑧 ∈ On ↔ {∅} ∈ On))
26253anbi3d 1213 . . . . . . . 8 (𝑧 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On)))
27 eleq2 2101 . . . . . . . . . 10 (𝑧 = {∅} → (𝑥𝑧𝑥 ∈ {∅}))
28 eleq1 2100 . . . . . . . . . 10 (𝑧 = {∅} → (𝑧𝑦 ↔ {∅} ∈ 𝑦))
2927, 28orbi12d 707 . . . . . . . . 9 (𝑧 = {∅} → ((𝑥𝑧𝑧𝑦) ↔ (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
3029imbi2d 219 . . . . . . . 8 (𝑧 = {∅} → ((𝑥𝑦 → (𝑥𝑧𝑧𝑦)) ↔ (𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦))))
3126, 30imbi12d 223 . . . . . . 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 4034 . . . . . . . . . . 11 ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦))))
3432, 33mpbi 133 . . . . . . . . . 10 ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦)))
3534simpri 106 . . . . . . . . 9 𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦))
36 epel 4029 . . . . . . . . . . . 12 (𝑥 E 𝑦𝑥𝑦)
37 epel 4029 . . . . . . . . . . . . 13 (𝑥 E 𝑧𝑥𝑧)
38 epel 4029 . . . . . . . . . . . . 13 (𝑧 E 𝑦𝑧𝑦)
3937, 38orbi12i 681 . . . . . . . . . . . 12 ((𝑥 E 𝑧𝑧 E 𝑦) ↔ (𝑥𝑧𝑧𝑦))
4036, 39imbi12i 228 . . . . . . . . . . 11 ((𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦)) ↔ (𝑥𝑦 → (𝑥𝑧𝑧𝑦)))
41402ralbii 2332 . . . . . . . . . 10 (∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥 E 𝑦 → (𝑥 E 𝑧𝑧 E 𝑦)) ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥𝑦 → (𝑥𝑧𝑧𝑦)))
4241ralbii 2330 . . . . . . . . 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 133 . . . . . . . 8 𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑥𝑦 → (𝑥𝑧𝑧𝑦))
4443rspec3 2409 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑥𝑦 → (𝑥𝑧𝑧𝑦)))
4524, 31, 44vtocl 2608 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ {∅} ∈ On) → (𝑥𝑦 → (𝑥 ∈ {∅} ∨ {∅} ∈ 𝑦)))
4616, 23, 45vtocl 2608 . . . . 5 ((𝑥 ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → (𝑥 ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
472, 15, 46vtocl 2608 . . . 4 (({𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ suc {𝑤 ∈ {∅} ∣ 𝜑} ∈ On ∧ {∅} ∈ On) → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑})))
481, 4, 8, 47mp3an 1232 . . 3 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}))
492elsn 3391 . . . . 5 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ↔ {𝑤 ∈ {∅} ∣ 𝜑} = ∅)
50 ordtriexmidlem2 4246 . . . . 5 ({𝑤 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
5149, 50sylbi 114 . . . 4 ({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
52 elirr 4266 . . . . . . 7 ¬ {∅} ∈ {∅}
53 elrabi 2695 . . . . . . 7 ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} ∈ {∅})
5452, 53mto 588 . . . . . 6 ¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑}
55 elsuci 4140 . . . . . . 7 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → ({∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
5655ord 643 . . . . . 6 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → (¬ {∅} ∈ {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑}))
5754, 56mpi 15 . . . . 5 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → {∅} = {𝑤 ∈ {∅} ∣ 𝜑})
58 0ex 3884 . . . . . . 7 ∅ ∈ V
59 biidd 161 . . . . . . 7 (𝑤 = ∅ → (𝜑𝜑))
6058, 59rabsnt 3445 . . . . . 6 ({𝑤 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
6160eqcoms 2043 . . . . 5 ({∅} = {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
6257, 61syl 14 . . . 4 ({∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑} → 𝜑)
6351, 62orim12i 676 . . 3 (({𝑤 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {∅} ∈ suc {𝑤 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑𝜑))
643, 48, 63mp2b 8 . 2 𝜑𝜑)
65 orcom 647 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
6664, 65mpbi 133 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  ∅c0 3224  {csn 3375   class class class wbr 3764   E cep 4024   Po wpo 4031   Or wor 4032  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  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  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-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-eprel 4026  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by: (None)
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