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 2536	. . . . . . . 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 2089	. . . . 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 2082	. . . . . . . 8 ⊢ (x = 𝑎 → (x ∈ ∪ y ↔ 𝑎 ∈ ∪ y))
12		suceq 4086	. . . . . . . . 9 ⊢ (x = 𝑎 → suc x = suc 𝑎)
13	12	eleq1d 2088	. . . . . . . 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 3561	. . . . . . . . 9 ⊢ (y = suc 𝑏 → ∪ y = ∪ suc 𝑏)
16	15	eleq2d 2089	. . . . . . . 8 ⊢ (y = suc 𝑏 → (𝑎 ∈ ∪ y ↔ 𝑎 ∈ ∪ suc 𝑏))
17		eleq2 2083	. . . . . . . 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 2638	. . . . . 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 2351	. 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 1374  ∀wral 2282  ∪ cuni 3552  Tr wtr 3826  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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-nul 3855  ax-pow 3899  ax-pr 3916  ax-un 4118

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-rab 2291  df-v 2535  df-dif 2895  df-un 2897  df-in 2899  df-ss 2906  df-nul 3200  df-pw 3334  df-sn 3354  df-pr 3355  df-uni 3553  df-tr 3827  df-iord 4050  df-on 4052  df-suc 4055

This theorem is referenced by: (None)