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Theorem onsucsssucexmid 4305
Description: The converse of onsucsssucr 4288 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
Hypothesis
Ref Expression
onsucsssucexmid.1 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦)
Assertion
Ref Expression
onsucsssucexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable groups:   𝜑,𝑥   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem onsucsssucexmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3090 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2 ordtriexmidlem 4298 . . . . . . 7 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 sseq1 3031 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
4 suceq 4192 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
54sseq1d 3037 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (suc 𝑥 ⊆ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
63, 5imbi12d 232 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})))
7 suc0 4201 . . . . . . . . . 10 suc ∅ = {∅}
8 0elon 4182 . . . . . . . . . . 11 ∅ ∈ On
98onsuci 4295 . . . . . . . . . 10 suc ∅ ∈ On
107, 9eqeltrri 2156 . . . . . . . . 9 {∅} ∈ On
11 p0ex 3986 . . . . . . . . . 10 {∅} ∈ V
12 eleq1 2145 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑦 ∈ On ↔ {∅} ∈ On))
1312anbi2d 452 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ {∅} ∈ On)))
14 sseq2 3032 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑥𝑦𝑥 ⊆ {∅}))
15 suceq 4192 . . . . . . . . . . . . 13 (𝑦 = {∅} → suc 𝑦 = suc {∅})
1615sseq2d 3038 . . . . . . . . . . . 12 (𝑦 = {∅} → (suc 𝑥 ⊆ suc 𝑦 ↔ suc 𝑥 ⊆ suc {∅}))
1714, 16imbi12d 232 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦) ↔ (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅})))
1813, 17imbi12d 232 . . . . . . . . . 10 (𝑦 = {∅} → (((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦)) ↔ ((𝑥 ∈ On ∧ {∅} ∈ On) → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))))
19 onsucsssucexmid.1 . . . . . . . . . . 11 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦)
2019rspec2 2456 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦))
2111, 18, 20vtocl 2664 . . . . . . . . 9 ((𝑥 ∈ On ∧ {∅} ∈ On) → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
2210, 21mpan2 416 . . . . . . . 8 (𝑥 ∈ On → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
236, 22vtoclga 2675 . . . . . . 7 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
242, 23ax-mp 7 . . . . . 6 ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
251, 24ax-mp 7 . . . . 5 suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}
2610onsuci 4295 . . . . . . 7 suc {∅} ∈ On
2726onordi 4216 . . . . . 6 Ord suc {∅}
28 ordelsuc 4284 . . . . . 6 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ Ord suc {∅}) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
292, 27, 28mp2an 417 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
3025, 29mpbir 144 . . . 4 {𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅}
31 elsucg 4194 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})))
322, 31ax-mp 7 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}))
3330, 32mpbi 143 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})
34 elsni 3440 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
35 ordtriexmidlem2 4299 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
3634, 35syl 14 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
37 0ex 3931 . . . . 5 ∅ ∈ V
38 biidd 170 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3937, 38rabsnt 3491 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
4036, 39orim12i 709 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑𝜑))
4133, 40ax-mp 7 . 2 𝜑𝜑)
42 orcom 680 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4341, 42mpbi 143 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  wral 2353  {crab 2357  wss 2984  c0 3269  {csn 3422  Ord word 4152  Oncon0 4153  suc csuc 4155
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158  df-suc 4161
This theorem is referenced by:  oawordriexmid  6162
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