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Theorem onsucsssucexmid 4449
Description: The converse of onsucsssucr 4432 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 3186 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2 ordtriexmidlem 4442 . . . . . . 7 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 sseq1 3124 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
4 suceq 4331 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
54sseq1d 3130 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (suc 𝑥 ⊆ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
63, 5imbi12d 233 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})))
7 suc0 4340 . . . . . . . . . 10 suc ∅ = {∅}
8 0elon 4321 . . . . . . . . . . 11 ∅ ∈ On
98onsuci 4439 . . . . . . . . . 10 suc ∅ ∈ On
107, 9eqeltrri 2214 . . . . . . . . 9 {∅} ∈ On
11 p0ex 4119 . . . . . . . . . 10 {∅} ∈ V
12 eleq1 2203 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑦 ∈ On ↔ {∅} ∈ On))
1312anbi2d 460 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ {∅} ∈ On)))
14 sseq2 3125 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑥𝑦𝑥 ⊆ {∅}))
15 suceq 4331 . . . . . . . . . . . . 13 (𝑦 = {∅} → suc 𝑦 = suc {∅})
1615sseq2d 3131 . . . . . . . . . . . 12 (𝑦 = {∅} → (suc 𝑥 ⊆ suc 𝑦 ↔ suc 𝑥 ⊆ suc {∅}))
1714, 16imbi12d 233 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦) ↔ (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅})))
1813, 17imbi12d 233 . . . . . . . . . 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 2524 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦))
2111, 18, 20vtocl 2743 . . . . . . . . 9 ((𝑥 ∈ On ∧ {∅} ∈ On) → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
2210, 21mpan2 422 . . . . . . . 8 (𝑥 ∈ On → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
236, 22vtoclga 2755 . . . . . . 7 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
242, 23ax-mp 5 . . . . . 6 ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
251, 24ax-mp 5 . . . . 5 suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}
2610onsuci 4439 . . . . . . 7 suc {∅} ∈ On
2726onordi 4355 . . . . . 6 Ord suc {∅}
28 ordelsuc 4428 . . . . . 6 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ Ord suc {∅}) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
292, 27, 28mp2an 423 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
3025, 29mpbir 145 . . . 4 {𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅}
31 elsucg 4333 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})))
322, 31ax-mp 5 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}))
3330, 32mpbi 144 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})
34 elsni 3549 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
35 ordtriexmidlem2 4443 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
3634, 35syl 14 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
37 0ex 4062 . . . . 5 ∅ ∈ V
38 biidd 171 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3937, 38rabsnt 3605 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
4036, 39orim12i 749 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑𝜑))
4133, 40ax-mp 5 . 2 𝜑𝜑)
42 orcom 718 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4341, 42mpbi 144 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698   = wceq 1332  wcel 1481  wral 2417  {crab 2421  wss 3075  c0 3367  {csn 3531  Ord word 4291  Oncon0 4292  suc csuc 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-tr 4034  df-iord 4295  df-on 4297  df-suc 4300
This theorem is referenced by:  oawordriexmid  6373
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