Theorem onsucelsucexmid 4453

Description: The converse of onsucelsucr 4432 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 4432 does hold, as seen at nnsucelsuc 6395. (Contributed by Jim Kingdon, 2-Aug-2019.)

Hypothesis

Ref	Expression
onsucelsucexmid.1	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → 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.

Step	Hyp	Ref	Expression
1		onsucelsucexmidlem1 4451	. . . 4 ⊢ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
2		0elon 4322	. . . . . 6 ⊢ ∅ ∈ On
3		onsucelsucexmidlem 4452	. . . . . 6 ⊢ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On
4	2, 3	pm3.2i 270	. . . . 5 ⊢ (∅ ∈ On ∧ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On)
5		onsucelsucexmid.1	. . . . 5 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)
6		eleq1 2203	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
7		suceq 4332	. . . . . . . 8 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
8	7	eleq1d 2209	. . . . . . 7 ⊢ (𝑥 = ∅ → (suc 𝑥 ∈ suc 𝑦 ↔ suc ∅ ∈ suc 𝑦))
9	6, 8	imbi12d 233	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦) ↔ (∅ ∈ 𝑦 → suc ∅ ∈ suc 𝑦)))
10		eleq2 2204	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (∅ ∈ 𝑦 ↔ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
11		suceq 4332	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc 𝑦 = suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
12	11	eleq2d 2210	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (suc ∅ ∈ suc 𝑦 ↔ suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
13	10, 12	imbi12d 233	. . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ((∅ ∈ 𝑦 → suc ∅ ∈ suc 𝑦) ↔ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})))
14	9, 13	rspc2va 2807	. . . . 5 ⊢ (((∅ ∈ On ∧ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)) → (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
15	4, 5, 14	mp2an 423	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
16	1, 15	ax-mp 5	. . 3 ⊢ suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
17		elsuci 4333	. . 3 ⊢ (suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∨ suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
18	16, 17	ax-mp 5	. 2 ⊢ (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∨ suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
19		suc0 4341	. . . . . 6 ⊢ suc ∅ = {∅}
20		p0ex 4120	. . . . . . 7 ⊢ {∅} ∈ V
21	20	prid2 3638	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
22	19, 21	eqeltri 2213	. . . . 5 ⊢ suc ∅ ∈ {∅, {∅}}
23		eqeq1 2147	. . . . . . 7 ⊢ (𝑧 = suc ∅ → (𝑧 = ∅ ↔ suc ∅ = ∅))
24	23	orbi1d 781	. . . . . 6 ⊢ (𝑧 = suc ∅ → ((𝑧 = ∅ ∨ 𝜑) ↔ (suc ∅ = ∅ ∨ 𝜑)))
25	24	elrab3 2845	. . . . 5 ⊢ (suc ∅ ∈ {∅, {∅}} → (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑)))
26	22, 25	ax-mp 5	. . . 4 ⊢ (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑))
27		0ex 4063	. . . . . . 7 ⊢ ∅ ∈ V
28		nsuceq0g 4348	. . . . . . 7 ⊢ (∅ ∈ V → suc ∅ ≠ ∅)
29	27, 28	ax-mp 5	. . . . . 6 ⊢ suc ∅ ≠ ∅
30		df-ne 2310	. . . . . 6 ⊢ (suc ∅ ≠ ∅ ↔ ¬ suc ∅ = ∅)
31	29, 30	mpbi 144	. . . . 5 ⊢ ¬ suc ∅ = ∅
32		pm2.53 712	. . . . 5 ⊢ ((suc ∅ = ∅ ∨ 𝜑) → (¬ suc ∅ = ∅ → 𝜑))
33	31, 32	mpi 15	. . . 4 ⊢ ((suc ∅ = ∅ ∨ 𝜑) → 𝜑)
34	26, 33	sylbi 120	. . 3 ⊢ (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → 𝜑)
35	19	eqeq1i 2148	. . . . 5 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ {∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
36	19	eqeq1i 2148	. . . . . . . 8 ⊢ (suc ∅ = ∅ ↔ {∅} = ∅)
37	31, 36	mtbi 660	. . . . . . 7 ⊢ ¬ {∅} = ∅
38	20	elsn 3548	. . . . . . 7 ⊢ ({∅} ∈ {∅} ↔ {∅} = ∅)
39	37, 38	mtbir 661	. . . . . 6 ⊢ ¬ {∅} ∈ {∅}
40		eleq2 2204	. . . . . 6 ⊢ ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ({∅} ∈ {∅} ↔ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
41	39, 40	mtbii 664	. . . . 5 ⊢ ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
42	35, 41	sylbi 120	. . . 4 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
43		olc 701	. . . . 5 ⊢ (𝜑 → ({∅} = ∅ ∨ 𝜑))
44		eqeq1 2147	. . . . . . . 8 ⊢ (𝑧 = {∅} → (𝑧 = ∅ ↔ {∅} = ∅))
45	44	orbi1d 781	. . . . . . 7 ⊢ (𝑧 = {∅} → ((𝑧 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
46	45	elrab3 2845	. . . . . 6 ⊢ ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
47	21, 46	ax-mp 5	. . . . 5 ⊢ ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
48	43, 47	sylibr 133	. . . 4 ⊢ (𝜑 → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
49	42, 48	nsyl 618	. . 3 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ 𝜑)
50	34, 49	orim12i 749	. 2 ⊢ ((suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∨ suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}) → (𝜑 ∨ ¬ 𝜑))
51	18, 50	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1332   ∈ wcel 1481   ≠ wne 2309  ∀wral 2417  {crab 2421  Vcvv 2689  ∅c0 3368  {csn 3532  {cpr 3533  Oncon0 4293  suc csuc 4295

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-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106

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-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301

This theorem is referenced by:  ordsucunielexmid  4454