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Theorem onsucsssucexmid 4252
 Description: The converse of onsucsssucr 4235 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 3025 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2 ordtriexmidlem 4245 . . . . . . 7 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 sseq1 2966 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
4 suceq 4139 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
54sseq1d 2972 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (suc 𝑥 ⊆ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
63, 5imbi12d 223 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})))
7 suc0 4148 . . . . . . . . . 10 suc ∅ = {∅}
8 0elon 4129 . . . . . . . . . . 11 ∅ ∈ On
98onsuci 4242 . . . . . . . . . 10 suc ∅ ∈ On
107, 9eqeltrri 2111 . . . . . . . . 9 {∅} ∈ On
11 p0ex 3939 . . . . . . . . . 10 {∅} ∈ V
12 eleq1 2100 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑦 ∈ On ↔ {∅} ∈ On))
1312anbi2d 437 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ {∅} ∈ On)))
14 sseq2 2967 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑥𝑦𝑥 ⊆ {∅}))
15 suceq 4139 . . . . . . . . . . . . 13 (𝑦 = {∅} → suc 𝑦 = suc {∅})
1615sseq2d 2973 . . . . . . . . . . . 12 (𝑦 = {∅} → (suc 𝑥 ⊆ suc 𝑦 ↔ suc 𝑥 ⊆ suc {∅}))
1714, 16imbi12d 223 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦) ↔ (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅})))
1813, 17imbi12d 223 . . . . . . . . . 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 2408 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦))
2111, 18, 20vtocl 2608 . . . . . . . . 9 ((𝑥 ∈ On ∧ {∅} ∈ On) → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
2210, 21mpan2 401 . . . . . . . 8 (𝑥 ∈ On → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
236, 22vtoclga 2619 . . . . . . 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 4242 . . . . . . 7 suc {∅} ∈ On
2726onordi 4163 . . . . . 6 Ord suc {∅}
28 ordelsuc 4231 . . . . . 6 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ Ord suc {∅}) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
292, 27, 28mp2an 402 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
3025, 29mpbir 134 . . . 4 {𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅}
31 elsucg 4141 . . . . 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 133 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})
34 elsni 3393 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
35 ordtriexmidlem2 4246 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
3634, 35syl 14 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
37 0ex 3884 . . . . 5 ∅ ∈ V
38 biidd 161 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3937, 38rabsnt 3445 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
4036, 39orim12i 676 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑𝜑))
4133, 40ax-mp 7 . 2 𝜑𝜑)
42 orcom 647 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4341, 42mpbi 133 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310   ⊆ wss 2917  ∅c0 3224  {csn 3375  Ord word 4099  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 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  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-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by: (None)
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