Theorem ordsucunielexmid 4629

Description: The converse of sucunielr 4608 (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 4472	. . . . . . . 8 ⊢ (𝑏 ∈ On → Ord 𝑏)
2		ordtr 4475	. . . . . . . 8 ⊢ (Ord 𝑏 → Tr 𝑏)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝑏 ∈ On → Tr 𝑏)
4		vex 2805	. . . . . . . 8 ⊢ 𝑏 ∈ V
5	4	unisuc 4510	. . . . . . 7 ⊢ (Tr 𝑏 ↔ ∪ suc 𝑏 = 𝑏)
6	3, 5	sylib 122	. . . . . 6 ⊢ (𝑏 ∈ On → ∪ suc 𝑏 = 𝑏)
7	6	eleq2d 2301	. . . . 5 ⊢ (𝑏 ∈ On → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
8	7	adantl 277	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
9		onsuc 4599	. . . . 5 ⊢ (𝑏 ∈ On → suc 𝑏 ∈ On)
10		ordsucunielexmid.1	. . . . . 6 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)
11		eleq1 2294	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ∪ 𝑦 ↔ 𝑎 ∈ ∪ 𝑦))
12		suceq 4499	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
13	12	eleq1d 2300	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (suc 𝑥 ∈ 𝑦 ↔ suc 𝑎 ∈ 𝑦))
14	11, 13	imbi12d 234	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦) ↔ (𝑎 ∈ ∪ 𝑦 → suc 𝑎 ∈ 𝑦)))
15		unieq 3902	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑏 → ∪ 𝑦 = ∪ suc 𝑏)
16	15	eleq2d 2301	. . . . . . . 8 ⊢ (𝑦 = suc 𝑏 → (𝑎 ∈ ∪ 𝑦 ↔ 𝑎 ∈ ∪ suc 𝑏))
17		eleq2 2295	. . . . . . . 8 ⊢ (𝑦 = suc 𝑏 → (suc 𝑎 ∈ 𝑦 ↔ suc 𝑎 ∈ suc 𝑏))
18	16, 17	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = suc 𝑏 → ((𝑎 ∈ ∪ 𝑦 → suc 𝑎 ∈ 𝑦) ↔ (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
19	14, 18	rspc2va 2924	. . . . . 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 425	. . . . 5 ⊢ ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
21	9, 20	sylan2 286	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
22	8, 21	sylbird 170	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏))
23	22	rgen2a 2586	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏)
24	23	onsucelsucexmid 4628	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∪ cuni 3893  Tr wtr 4187  Ord word 4459  Oncon0 4460  suc csuc 4462

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by: (None)