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Theorem onsucsssucexmid 4522
Description: The converse of onsucsssucr 4504 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 3240 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2 ordtriexmidlem 4514 . . . . . . 7 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3 sseq1 3178 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
4 suceq 4398 . . . . . . . . . 10 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
54sseq1d 3184 . . . . . . . . 9 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (suc 𝑥 ⊆ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
63, 5imbi12d 234 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})))
7 suc0 4407 . . . . . . . . . 10 suc ∅ = {∅}
8 0elon 4388 . . . . . . . . . . 11 ∅ ∈ On
98onsuci 4511 . . . . . . . . . 10 suc ∅ ∈ On
107, 9eqeltrri 2251 . . . . . . . . 9 {∅} ∈ On
11 p0ex 4185 . . . . . . . . . 10 {∅} ∈ V
12 eleq1 2240 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑦 ∈ On ↔ {∅} ∈ On))
1312anbi2d 464 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ {∅} ∈ On)))
14 sseq2 3179 . . . . . . . . . . . 12 (𝑦 = {∅} → (𝑥𝑦𝑥 ⊆ {∅}))
15 suceq 4398 . . . . . . . . . . . . 13 (𝑦 = {∅} → suc 𝑦 = suc {∅})
1615sseq2d 3185 . . . . . . . . . . . 12 (𝑦 = {∅} → (suc 𝑥 ⊆ suc 𝑦 ↔ suc 𝑥 ⊆ suc {∅}))
1714, 16imbi12d 234 . . . . . . . . . . 11 (𝑦 = {∅} → ((𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦) ↔ (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅})))
1813, 17imbi12d 234 . . . . . . . . . 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 2566 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦))
2111, 18, 20vtocl 2791 . . . . . . . . 9 ((𝑥 ∈ On ∧ {∅} ∈ On) → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
2210, 21mpan2 425 . . . . . . . 8 (𝑥 ∈ On → (𝑥 ⊆ {∅} → suc 𝑥 ⊆ suc {∅}))
236, 22vtoclga 2803 . . . . . . 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 4511 . . . . . . 7 suc {∅} ∈ On
2726onordi 4422 . . . . . 6 Ord suc {∅}
28 ordelsuc 4500 . . . . . 6 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ On ∧ Ord suc {∅}) → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅}))
292, 27, 28mp2an 426 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅} ↔ suc {𝑧 ∈ {∅} ∣ 𝜑} ⊆ suc {∅})
3025, 29mpbir 146 . . . 4 {𝑧 ∈ {∅} ∣ 𝜑} ∈ suc {∅}
31 elsucg 4400 . . . . 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 145 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅})
34 elsni 3609 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
35 ordtriexmidlem2 4515 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
3634, 35syl 14 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} → ¬ 𝜑)
37 0ex 4127 . . . . 5 ∅ ∈ V
38 biidd 172 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3937, 38rabsnt 3666 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = {∅} → 𝜑)
4036, 39orim12i 759 . . 3 (({𝑧 ∈ {∅} ∣ 𝜑} ∈ {∅} ∨ {𝑧 ∈ {∅} ∣ 𝜑} = {∅}) → (¬ 𝜑𝜑))
4133, 40ax-mp 5 . 2 𝜑𝜑)
42 orcom 728 . 2 ((¬ 𝜑𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
4341, 42mpbi 145 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148  wral 2455  {crab 2459  wss 3129  c0 3422  {csn 3591  Ord word 4358  Oncon0 4359  suc csuc 4361
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-tr 4099  df-iord 4362  df-on 4364  df-suc 4367
This theorem is referenced by:  oawordriexmid  6464
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