Theorem onsucelsucexmid 4507

Description: The converse of onsucelsucr 4485 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 4485 does hold, as seen at nnsucelsuc 6459. (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 4505	. . . 4 ⊢ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
2		0elon 4370	. . . . . 6 ⊢ ∅ ∈ On
3		onsucelsucexmidlem 4506	. . . . . 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 2229	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
7		suceq 4380	. . . . . . . 8 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
8	7	eleq1d 2235	. . . . . . 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 2230	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (∅ ∈ 𝑦 ↔ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
11		suceq 4380	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc 𝑦 = suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
12	11	eleq2d 2236	. . . . . . 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 2844	. . . . 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 4381	. . 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 4389	. . . . . 6 ⊢ suc ∅ = {∅}
20		p0ex 4167	. . . . . . 7 ⊢ {∅} ∈ V
21	20	prid2 3683	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
22	19, 21	eqeltri 2239	. . . . 5 ⊢ suc ∅ ∈ {∅, {∅}}
23		eqeq1 2172	. . . . . . 7 ⊢ (𝑧 = suc ∅ → (𝑧 = ∅ ↔ suc ∅ = ∅))
24	23	orbi1d 781	. . . . . 6 ⊢ (𝑧 = suc ∅ → ((𝑧 = ∅ ∨ 𝜑) ↔ (suc ∅ = ∅ ∨ 𝜑)))
25	24	elrab3 2883	. . . . 5 ⊢ (suc ∅ ∈ {∅, {∅}} → (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑)))
26	22, 25	ax-mp 5	. . . 4 ⊢ (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑))
27		0ex 4109	. . . . . . 7 ⊢ ∅ ∈ V
28		nsuceq0g 4396	. . . . . . 7 ⊢ (∅ ∈ V → suc ∅ ≠ ∅)
29	27, 28	ax-mp 5	. . . . . 6 ⊢ suc ∅ ≠ ∅
30		df-ne 2337	. . . . . 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 2173	. . . . 5 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ {∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
36	19	eqeq1i 2173	. . . . . . . 8 ⊢ (suc ∅ = ∅ ↔ {∅} = ∅)
37	31, 36	mtbi 660	. . . . . . 7 ⊢ ¬ {∅} = ∅
38	20	elsn 3592	. . . . . . 7 ⊢ ({∅} ∈ {∅} ↔ {∅} = ∅)
39	37, 38	mtbir 661	. . . . . 6 ⊢ ¬ {∅} ∈ {∅}
40		eleq2 2230	. . . . . 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 2172	. . . . . . . 8 ⊢ (𝑧 = {∅} → (𝑧 = ∅ ↔ {∅} = ∅))
45	44	orbi1d 781	. . . . . . 7 ⊢ (𝑧 = {∅} → ((𝑧 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
46	45	elrab3 2883	. . . . . 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 1343   ∈ wcel 2136   ≠ wne 2336  ∀wral 2444  {crab 2448  Vcvv 2726  ∅c0 3409  {csn 3576  {cpr 3577  Oncon0 4341  suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  ordsucunielexmid  4508