Theorem ordsucunielexmid 4274

Description: The converse of sucunielr 4254 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)

Hypothesis

Ref	Expression
ordsucunielexmid.1	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)

Assertion

Ref	Expression
ordsucunielexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ordsucunielexmid

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eloni 4130	. . . . . . . 8 ⊢ (𝑏 ∈ On → Ord 𝑏)
2		ordtr 4133	. . . . . . . 8 ⊢ (Ord 𝑏 → Tr 𝑏)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝑏 ∈ On → Tr 𝑏)
4		vex 2604	. . . . . . . 8 ⊢ 𝑏 ∈ V
5	4	unisuc 4168	. . . . . . 7 ⊢ (Tr 𝑏 ↔ ∪ suc 𝑏 = 𝑏)
6	3, 5	sylib 120	. . . . . 6 ⊢ (𝑏 ∈ On → ∪ suc 𝑏 = 𝑏)
7	6	eleq2d 2148	. . . . 5 ⊢ (𝑏 ∈ On → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
8	7	adantl 271	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
9		suceloni 4245	. . . . 5 ⊢ (𝑏 ∈ On → suc 𝑏 ∈ On)
10		ordsucunielexmid.1	. . . . . 6 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)
11		eleq1 2141	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ∪ 𝑦 ↔ 𝑎 ∈ ∪ 𝑦))
12		suceq 4157	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
13	12	eleq1d 2147	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (suc 𝑥 ∈ 𝑦 ↔ suc 𝑎 ∈ 𝑦))
14	11, 13	imbi12d 232	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦) ↔ (𝑎 ∈ ∪ 𝑦 → suc 𝑎 ∈ 𝑦)))
15		unieq 3610	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑏 → ∪ 𝑦 = ∪ suc 𝑏)
16	15	eleq2d 2148	. . . . . . . 8 ⊢ (𝑦 = suc 𝑏 → (𝑎 ∈ ∪ 𝑦 ↔ 𝑎 ∈ ∪ suc 𝑏))
17		eleq2 2142	. . . . . . . 8 ⊢ (𝑦 = suc 𝑏 → (suc 𝑎 ∈ 𝑦 ↔ suc 𝑎 ∈ suc 𝑏))
18	16, 17	imbi12d 232	. . . . . . 7 ⊢ (𝑦 = suc 𝑏 → ((𝑎 ∈ ∪ 𝑦 → suc 𝑎 ∈ 𝑦) ↔ (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
19	14, 18	rspc2va 2714	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ suc 𝑏 ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
20	10, 19	mpan2 415	. . . . 5 ⊢ ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
21	9, 20	sylan2 280	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
22	8, 21	sylbird 168	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏))
23	22	rgen2a 2417	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏)
24	23	onsucelsucexmid 4273	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∪ cuni 3601  Tr wtr 3875  Ord word 4117  Oncon0 4118  suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by: (None)