Theorem onsucelsucexmid 4599

Description: The converse of onsucelsucr 4577 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 4577 does hold, as seen at nnsucelsuc 6607. (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 4597	. . . 4 ⊢ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}
2		0elon 4460	. . . . . 6 ⊢ ∅ ∈ On
3		onsucelsucexmidlem 4598	. . . . . 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 2272	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
7		suceq 4470	. . . . . . . 8 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
8	7	eleq1d 2278	. . . . . . 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 2273	. . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → (∅ ∈ 𝑦 ↔ ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
11		suceq 4470	. . . . . . . 8 ⊢ (𝑦 = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → suc 𝑦 = suc {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
12	11	eleq2d 2279	. . . . . . 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 2901	. . . . 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 4471	. . 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 4479	. . . . . 6 ⊢ suc ∅ = {∅}
20		p0ex 4251	. . . . . . 7 ⊢ {∅} ∈ V
21	20	prid2 3753	. . . . . 6 ⊢ {∅} ∈ {∅, {∅}}
22	19, 21	eqeltri 2282	. . . . 5 ⊢ suc ∅ ∈ {∅, {∅}}
23		eqeq1 2216	. . . . . . 7 ⊢ (𝑧 = suc ∅ → (𝑧 = ∅ ↔ suc ∅ = ∅))
24	23	orbi1d 795	. . . . . 6 ⊢ (𝑧 = suc ∅ → ((𝑧 = ∅ ∨ 𝜑) ↔ (suc ∅ = ∅ ∨ 𝜑)))
25	24	elrab3 2940	. . . . 5 ⊢ (suc ∅ ∈ {∅, {∅}} → (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑)))
26	22, 25	ax-mp 5	. . . 4 ⊢ (suc ∅ ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ (suc ∅ = ∅ ∨ 𝜑))
27		0ex 4190	. . . . . . 7 ⊢ ∅ ∈ V
28		nsuceq0g 4486	. . . . . . 7 ⊢ (∅ ∈ V → suc ∅ ≠ ∅)
29	27, 28	ax-mp 5	. . . . . 6 ⊢ suc ∅ ≠ ∅
30		df-ne 2381	. . . . . 6 ⊢ (suc ∅ ≠ ∅ ↔ ¬ suc ∅ = ∅)
31	29, 30	mpbi 145	. . . . 5 ⊢ ¬ suc ∅ = ∅
32		pm2.53 726	. . . . 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 2217	. . . . 5 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} ↔ {∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
36	19	eqeq1i 2217	. . . . . . . 8 ⊢ (suc ∅ = ∅ ↔ {∅} = ∅)
37	31, 36	mtbi 674	. . . . . . 7 ⊢ ¬ {∅} = ∅
38	20	elsn 3662	. . . . . . 7 ⊢ ({∅} ∈ {∅} ↔ {∅} = ∅)
39	37, 38	mtbir 675	. . . . . 6 ⊢ ¬ {∅} ∈ {∅}
40		eleq2 2273	. . . . . 6 ⊢ ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ({∅} ∈ {∅} ↔ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)}))
41	39, 40	mtbii 678	. . . . 5 ⊢ ({∅} = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
42	35, 41	sylbi 121	. . . 4 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ {∅} ∈ {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)})
43		olc 715	. . . . 5 ⊢ (𝜑 → ({∅} = ∅ ∨ 𝜑))
44		eqeq1 2216	. . . . . . . 8 ⊢ (𝑧 = {∅} → (𝑧 = ∅ ↔ {∅} = ∅))
45	44	orbi1d 795	. . . . . . 7 ⊢ (𝑧 = {∅} → ((𝑧 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
46	45	elrab3 2940	. . . . . 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 631	. . 3 ⊢ (suc ∅ = {𝑧 ∈ {∅, {∅}} ∣ (𝑧 = ∅ ∨ 𝜑)} → ¬ 𝜑)
50	34, 49	orim12i 763	. 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 712   = wceq 1375   ∈ wcel 2180   ≠ wne 2380  ∀wral 2488  {crab 2492  Vcvv 2779  ∅c0 3471  {csn 3646  {cpr 3647  Oncon0 4431  suc csuc 4433

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-uni 3868  df-tr 4162  df-iord 4434  df-on 4436  df-suc 4439

This theorem is referenced by:  ordsucunielexmid  4600