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Theorem onsucelsucexmid 4541
Description: The converse of onsucelsucr 4519 implies excluded middle. On the other hand, if 𝑦 is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4519 does hold, as seen at nnsucelsuc 6505. (Contributed by Jim Kingdon, 2-Aug-2019.)
Hypothesis
Ref Expression
onsucelsucexmid.1 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ∈ suc 𝑦)
Assertion
Ref Expression
onsucelsucexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem onsucelsucexmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 onsucelsucexmidlem1 4539 . . . 4 ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
2 0elon 4404 . . . . . 6 ∅ ∈ On
3 onsucelsucexmidlem 4540 . . . . . 6 {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On
42, 3pm3.2i 272 . . . . 5 (∅ ∈ On ∧ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On)
5 onsucelsucexmid.1 . . . . 5 𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ∈ suc 𝑦)
6 eleq1 2250 . . . . . . 7 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
7 suceq 4414 . . . . . . . 8 (𝑥 = ∅ → suc 𝑥 = suc ∅)
87eleq1d 2256 . . . . . . 7 (𝑥 = ∅ → (suc 𝑥 ∈ suc 𝑦 ↔ suc ∅ ∈ suc 𝑦))
96, 8imbi12d 234 . . . . . 6 (𝑥 = ∅ → ((𝑥𝑦 → suc 𝑥 ∈ suc 𝑦) ↔ (∅ ∈ 𝑦 → suc ∅ ∈ suc 𝑦)))
10 eleq2 2251 . . . . . . 7 (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (∅ ∈ 𝑦 ↔ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
11 suceq 4414 . . . . . . . 8 (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc 𝑦 = suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
1211eleq2d 2257 . . . . . . 7 (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (suc ∅ ∈ suc 𝑦 ↔ suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
1310, 12imbi12d 234 . . . . . 6 (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ((∅ ∈ 𝑦 → suc ∅ ∈ suc 𝑦) ↔ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})))
149, 13rspc2va 2867 . . . . 5 (((∅ ∈ On ∧ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ∈ suc 𝑦)) → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
154, 5, 14mp2an 426 . . . 4 (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
161, 15ax-mp 5 . . 3 suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
17 elsuci 4415 . . 3 (suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∨ suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
1816, 17ax-mp 5 . 2 (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∨ suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
19 suc0 4423 . . . . . 6 suc ∅ = {∅}
20 p0ex 4200 . . . . . . 7 {∅} ∈ V
2120prid2 3711 . . . . . 6 {∅} ∈ {∅, {∅}}
2219, 21eqeltri 2260 . . . . 5 suc ∅ ∈ {∅, {∅}}
23 eqeq1 2194 . . . . . . 7 (𝑧 = suc ∅ → (𝑧 = ∅ ↔ suc ∅ = ∅))
2423orbi1d 792 . . . . . 6 (𝑧 = suc ∅ → ((𝑧 = ∅ ∨ 𝜑) ↔ (suc ∅ = ∅ ∨ 𝜑)))
2524elrab3 2906 . . . . 5 (suc ∅ ∈ {∅, {∅}} → (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑)))
2622, 25ax-mp 5 . . . 4 (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑))
27 0ex 4142 . . . . . . 7 ∅ ∈ V
28 nsuceq0g 4430 . . . . . . 7 (∅ ∈ V → suc ∅ ≠ ∅)
2927, 28ax-mp 5 . . . . . 6 suc ∅ ≠ ∅
30 df-ne 2358 . . . . . 6 (suc ∅ ≠ ∅ ↔ ¬ suc ∅ = ∅)
3129, 30mpbi 145 . . . . 5 ¬ suc ∅ = ∅
32 pm2.53 723 . . . . 5 ((suc ∅ = ∅ ∨ 𝜑) → (¬ suc ∅ = ∅ → 𝜑))
3331, 32mpi 15 . . . 4 ((suc ∅ = ∅ ∨ 𝜑) → 𝜑)
3426, 33sylbi 121 . . 3 (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → 𝜑)
3519eqeq1i 2195 . . . . 5 (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ {∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
3619eqeq1i 2195 . . . . . . . 8 (suc ∅ = ∅ ↔ {∅} = ∅)
3731, 36mtbi 671 . . . . . . 7 ¬ {∅} = ∅
3820elsn 3620 . . . . . . 7 ({∅} ∈ {∅} ↔ {∅} = ∅)
3937, 38mtbir 672 . . . . . 6 ¬ {∅} ∈ {∅}
40 eleq2 2251 . . . . . 6 ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ({∅} ∈ {∅} ↔ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
4139, 40mtbii 675 . . . . 5 ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
4235, 41sylbi 121 . . . 4 (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
43 olc 712 . . . . 5 (𝜑 → ({∅} = ∅ ∨ 𝜑))
44 eqeq1 2194 . . . . . . . 8 (𝑧 = {∅} → (𝑧 = ∅ ↔ {∅} = ∅))
4544orbi1d 792 . . . . . . 7 (𝑧 = {∅} → ((𝑧 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
4645elrab3 2906 . . . . . 6 ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
4721, 46ax-mp 5 . . . . 5 ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
4843, 47sylibr 134 . . . 4 (𝜑 → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
4942, 48nsyl 629 . . 3 (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ 𝜑)
5034, 49orim12i 760 . 2 ((suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∨ suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}) → (𝜑 ∨ ¬ 𝜑))
5118, 50ax-mp 5 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709   = wceq 1363  wcel 2158  wne 2357  wral 2465  {crab 2469  Vcvv 2749  c0 3434  {csn 3604  {cpr 3605  Oncon0 4375  suc csuc 4377
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by:  ordsucunielexmid  4542
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