Theorem ordsucunielexmid 4215

Description: The converse of sucunielr 4200 (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 4077	. . . . . . . 8 ⊢ (𝑏 ∈ On → Ord 𝑏)
2		ordtr 4080	. . . . . . . 8 ⊢ (Ord 𝑏 → Tr 𝑏)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝑏 ∈ On → Tr 𝑏)
4		vex 2554	. . . . . . . 8 ⊢ 𝑏 ∈ V
5	4	unisuc 4115	. . . . . . 7 ⊢ (Tr 𝑏 ↔ ∪ suc 𝑏 = 𝑏)
6	3, 5	sylib 127	. . . . . 6 ⊢ (𝑏 ∈ On → ∪ suc 𝑏 = 𝑏)
7	6	eleq2d 2104	. . . . 5 ⊢ (𝑏 ∈ On → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
8	7	adantl 262	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
9		suceloni 4192	. . . . 5 ⊢ (𝑏 ∈ On → suc 𝑏 ∈ On)
10		ordsucunielexmid.1	. . . . . 6 ⊢ ∀x ∈ On ∀y ∈ On (x ∈ ∪ y → suc x ∈ y)
11		eleq1 2097	. . . . . . . 8 ⊢ (x = 𝑎 → (x ∈ ∪ y ↔ 𝑎 ∈ ∪ y))
12		suceq 4104	. . . . . . . . 9 ⊢ (x = 𝑎 → suc x = suc 𝑎)
13	12	eleq1d 2103	. . . . . . . 8 ⊢ (x = 𝑎 → (suc x ∈ y ↔ suc 𝑎 ∈ y))
14	11, 13	imbi12d 223	. . . . . . 7 ⊢ (x = 𝑎 → ((x ∈ ∪ y → suc x ∈ y) ↔ (𝑎 ∈ ∪ y → suc 𝑎 ∈ y)))
15		unieq 3579	. . . . . . . . 9 ⊢ (y = suc 𝑏 → ∪ y = ∪ suc 𝑏)
16	15	eleq2d 2104	. . . . . . . 8 ⊢ (y = suc 𝑏 → (𝑎 ∈ ∪ y ↔ 𝑎 ∈ ∪ suc 𝑏))
17		eleq2 2098	. . . . . . . 8 ⊢ (y = suc 𝑏 → (suc 𝑎 ∈ y ↔ suc 𝑎 ∈ suc 𝑏))
18	16, 17	imbi12d 223	. . . . . . 7 ⊢ (y = suc 𝑏 → ((𝑎 ∈ ∪ y → suc 𝑎 ∈ y) ↔ (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
19	14, 18	rspc2va 2657	. . . . . 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 401	. . . . 5 ⊢ ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
21	9, 20	sylan2 270	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
22	8, 21	sylbird 159	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏))
23	22	rgen2a 2369	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏)
24	23	onsucelsucexmid 4214	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∪ cuni 3570  Tr wtr 3844  Ord word 4064  Oncon0 4065  suc csuc 4067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-uni 3571  df-tr 3845  df-iord 4068  df-on 4070  df-suc 4073

This theorem is referenced by: (None)