Theorem onsucelsucexmid 4544

Description: The converse of onsucelsucr 4522 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 4522 does hold, as seen at nnsucelsuc 6511. (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 4542	. . . 4 ⊢ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
2		0elon 4407	. . . . . 6 ⊢ ∅ ∈ On
3		onsucelsucexmidlem 4543	. . . . . 6 ⊢ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On
4	2, 3	pm3.2i 272	. . . . 5 ⊢ (∅ ∈ On ∧ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ∈ On)
5		onsucelsucexmid.1	. . . . 5 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)
6		eleq1 2252	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
7		suceq 4417	. . . . . . . 8 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
8	7	eleq1d 2258	. . . . . . 7 ⊢ (𝑥 = ∅ → (suc 𝑥 ∈ suc 𝑦 ↔ suc ∅ ∈ suc 𝑦))
9	6, 8	imbi12d 234	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦) ↔ (∅ ∈ 𝑦 → suc ∅ ∈ suc 𝑦)))
10		eleq2 2253	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (∅ ∈ 𝑦 ↔ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
11		suceq 4417	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc 𝑦 = suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
12	11	eleq2d 2259	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (suc ∅ ∈ suc 𝑦 ↔ suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
13	10, 12	imbi12d 234	. . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ((∅ ∈ 𝑦 → suc ∅ ∈ suc 𝑦) ↔ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})))
14	9, 13	rspc2va 2870	. . . . 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 426	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
16	1, 15	ax-mp 5	. . 3 ⊢ suc ∅ ∈ suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
17		elsuci 4418	. . 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 4426	. . . . . 6 ⊢ suc ∅ = {∅}
20		p0ex 4203	. . . . . . 7 ⊢ {∅} ∈ V
21	20	prid2 3714	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
22	19, 21	eqeltri 2262	. . . . 5 ⊢ suc ∅ ∈ {∅, {∅}}
23		eqeq1 2196	. . . . . . 7 ⊢ (𝑧 = suc ∅ → (𝑧 = ∅ ↔ suc ∅ = ∅))
24	23	orbi1d 792	. . . . . 6 ⊢ (𝑧 = suc ∅ → ((𝑧 = ∅ ∨ 𝜑) ↔ (suc ∅ = ∅ ∨ 𝜑)))
25	24	elrab3 2909	. . . . 5 ⊢ (suc ∅ ∈ {∅, {∅}} → (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑)))
26	22, 25	ax-mp 5	. . . 4 ⊢ (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑))
27		0ex 4145	. . . . . . 7 ⊢ ∅ ∈ V
28		nsuceq0g 4433	. . . . . . 7 ⊢ (∅ ∈ V → suc ∅ ≠ ∅)
29	27, 28	ax-mp 5	. . . . . 6 ⊢ suc ∅ ≠ ∅
30		df-ne 2361	. . . . . 6 ⊢ (suc ∅ ≠ ∅ ↔ ¬ suc ∅ = ∅)
31	29, 30	mpbi 145	. . . . 5 ⊢ ¬ suc ∅ = ∅
32		pm2.53 723	. . . . 5 ⊢ ((suc ∅ = ∅ ∨ 𝜑) → (¬ suc ∅ = ∅ → 𝜑))
33	31, 32	mpi 15	. . . 4 ⊢ ((suc ∅ = ∅ ∨ 𝜑) → 𝜑)
34	26, 33	sylbi 121	. . 3 ⊢ (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → 𝜑)
35	19	eqeq1i 2197	. . . . 5 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ {∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
36	19	eqeq1i 2197	. . . . . . . 8 ⊢ (suc ∅ = ∅ ↔ {∅} = ∅)
37	31, 36	mtbi 671	. . . . . . 7 ⊢ ¬ {∅} = ∅
38	20	elsn 3623	. . . . . . 7 ⊢ ({∅} ∈ {∅} ↔ {∅} = ∅)
39	37, 38	mtbir 672	. . . . . 6 ⊢ ¬ {∅} ∈ {∅}
40		eleq2 2253	. . . . . 6 ⊢ ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ({∅} ∈ {∅} ↔ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
41	39, 40	mtbii 675	. . . . 5 ⊢ ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
42	35, 41	sylbi 121	. . . 4 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
43		olc 712	. . . . 5 ⊢ (𝜑 → ({∅} = ∅ ∨ 𝜑))
44		eqeq1 2196	. . . . . . . 8 ⊢ (𝑧 = {∅} → (𝑧 = ∅ ↔ {∅} = ∅))
45	44	orbi1d 792	. . . . . . 7 ⊢ (𝑧 = {∅} → ((𝑧 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
46	45	elrab3 2909	. . . . . 6 ⊢ ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
47	21, 46	ax-mp 5	. . . . 5 ⊢ ({∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
48	43, 47	sylibr 134	. . . 4 ⊢ (𝜑 → {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
49	42, 48	nsyl 629	. . 3 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ 𝜑)
50	34, 49	orim12i 760	. 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 104   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2160   ≠ wne 2360  ∀wral 2468  {crab 2472  Vcvv 2752  ∅c0 3437  {csn 3607  {cpr 3608  Oncon0 4378  suc csuc 4380

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386

This theorem is referenced by:  ordsucunielexmid  4545