Theorem ordsucunielexmid 4198

Description: The converse of sucunielr 4183 (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 4059	. . . . . . . 8 ⊢ (𝑏 ∈ On → Ord 𝑏)
2		ordtr 4062	. . . . . . . 8 ⊢ (Ord 𝑏 → Tr 𝑏)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝑏 ∈ On → Tr 𝑏)
4		vex 2537	. . . . . . . 8 ⊢ 𝑏 ∈ V
5	4	unisuc 4097	. . . . . . 7 ⊢ (Tr 𝑏 ↔ ∪ suc 𝑏 = 𝑏)
6	3, 5	sylib 127	. . . . . 6 ⊢ (𝑏 ∈ On → ∪ suc 𝑏 = 𝑏)
7	6	eleq2d 2090	. . . . 5 ⊢ (𝑏 ∈ On → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
8	7	adantl 262	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
9		suceloni 4175	. . . . 5 ⊢ (𝑏 ∈ On → suc 𝑏 ∈ On)
10		ordsucunielexmid.1	. . . . . 6 ⊢ ∀x ∈ On ∀y ∈ On (x ∈ ∪ y → suc x ∈ y)
11		eleq1 2083	. . . . . . . 8 ⊢ (x = 𝑎 → (x ∈ ∪ y ↔ 𝑎 ∈ ∪ y))
12		suceq 4086	. . . . . . . . 9 ⊢ (x = 𝑎 → suc x = suc 𝑎)
13	12	eleq1d 2089	. . . . . . . 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 3562	. . . . . . . . 9 ⊢ (y = suc 𝑏 → ∪ y = ∪ suc 𝑏)
16	15	eleq2d 2090	. . . . . . . 8 ⊢ (y = suc 𝑏 → (𝑎 ∈ ∪ y ↔ 𝑎 ∈ ∪ suc 𝑏))
17		eleq2 2084	. . . . . . . 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 2639	. . . . . 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 403	. . . . 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 2352	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏)
24	23	onsucelsucexmid 4197	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1375  ∀wral 2283  ∪ cuni 3553  Tr wtr 3827  Ord word 4046  Oncon0 4047  suc csuc 4049

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-nul 3856  ax-pow 3900  ax-pr 3917  ax-un 4118

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ne 2189  df-ral 2288  df-rex 2289  df-rab 2292  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055

This theorem is referenced by: (None)