Theorem iundisj2 25506

Description: A disjoint union is disjoint. (Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
iundisj.1	⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)

Assertion

Ref	Expression
iundisj2	⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Distinct variable groups:   𝑘,𝑛   𝐴,𝑘   𝐵,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝐵(𝑘)

Proof of Theorem iundisj2

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

Step	Hyp	Ref	Expression
1		tru 1545	. . . 4 ⊢ ⊤
2		eqeq12 2753	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
3		csbeq1 3852	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
4		csbeq1 3852	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	3, 4	ineqan12d 4174	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
6	5	eqeq1d 2738	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅ ↔ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
7	2, 6	orbi12d 918	. . . . 5 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 = 𝑏 ∨ (⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅) ↔ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)))
8		eqeq12 2753	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑦 = 𝑥))
9		equcom 2019	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
10	8, 9	bitrdi 287	. . . . . 6 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
11		csbeq1 3852	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
12		csbeq1 3852	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
13	11, 12	ineqan12d 4174	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = (⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
14		incom 4161	. . . . . . . 8 ⊢ (⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
15	13, 14	eqtrdi 2787	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
16	15	eqeq1d 2738	. . . . . 6 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ((⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅ ↔ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
17	10, 16	orbi12d 918	. . . . 5 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ((𝑎 = 𝑏 ∨ (⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅) ↔ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)))
18		nnssre 12149	. . . . . 6 ⊢ ℕ ⊆ ℝ
19	18	a1i 11	. . . . 5 ⊢ (⊤ → ℕ ⊆ ℝ)
20		biidd 262	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → ((𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅) ↔ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)))
21		nesym 2988	. . . . . . . 8 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 = 𝑦)
22		nnre 12152	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
23		nnre 12152	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
24		id 22	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑦)
25		leltne 11222	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
26	22, 23, 24, 25	syl3an 1160	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
27		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
28		nfcsb1v 3873	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑥 / 𝑛⦌𝐴
29		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑥)𝐵
30	28, 29	nfdif 4081	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
31		csbeq1a 3863	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → 𝐴 = ⦋𝑥 / 𝑛⦌𝐴)
32		oveq2 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑥 → (1..^𝑛) = (1..^𝑥))
33	32	iuneq1d 4974	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑥)𝐵)
34	31, 33	difeq12d 4079	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵))
35	27, 30, 34	csbief 3883	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
36		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
37		nfcsb1v 3873	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴
38		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑦)𝐵
39	37, 38	nfdif 4081	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
40		csbeq1a 3863	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴)
41		oveq2 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → (1..^𝑛) = (1..^𝑦))
42	41	iuneq1d 4974	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑦)𝐵)
43	40, 42	difeq12d 4079	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵))
44	36, 39, 43	csbief 3883	. . . . . . . . . . . . . 14 ⊢ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
45	35, 44	ineq12i 4170	. . . . . . . . . . . . 13 ⊢ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ((⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵) ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵))
46		simp1 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℕ)
47		nnuz 12790	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ = (ℤ≥‘1)
48	46, 47	eleqtrdi 2846	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (ℤ≥‘1))
49		simp2 1137	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℕ)
50	49	nnzd 12514	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℤ)
51		simp3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
52		elfzo2 13578	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1..^𝑦) ↔ (𝑥 ∈ (ℤ≥‘1) ∧ 𝑦 ∈ ℤ ∧ 𝑥 < 𝑦))
53	48, 50, 51, 52	syl3anbrc 1344	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1..^𝑦))
54		nfcv 2898	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝑘
55		nfcv 2898	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝐵
56		iundisj.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)
57	54, 55, 56	csbhypf 3877	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
58	57	equcoms 2021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
59	58	eqcomd 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑛⦌𝐴)
60	59	ssiun2s 5004	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1..^𝑦) → ⦋𝑥 / 𝑛⦌𝐴 ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
61	53, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → ⦋𝑥 / 𝑛⦌𝐴 ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
62	61	ssdifssd 4099	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵) ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
63	62	ssrind 4196	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → ((⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵) ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) ⊆ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)))
64	45, 63	eqsstrid 3972	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) ⊆ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)))
65		disjdif 4424	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) = ∅
66		sseq0 4355	. . . . . . . . . . . 12 ⊢ (((⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) ⊆ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) ∧ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) = ∅) → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
67	64, 65, 66	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
68	67	3expia 1121	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
69	68	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
70	26, 69	sylbird 260	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑦 ≠ 𝑥 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
71	21, 70	biimtrrid 243	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (¬ 𝑥 = 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
72	71	orrd 863	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
73	72	adantl 481	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦)) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
74	7, 17, 19, 20, 73	wlogle 11670	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
75	1, 74	mpan 690	. . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
76	75	rgen2 3176	. 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
77		disjors 5081	. 2 ⊢ (Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
78	76, 77	mpbir 231	1 ⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ⦋csb 3849   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  ∪ ciun 4946  Disj wdisj 5065   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  ℝcr 11025  1c1 11027   < clt 11166   ≤ cle 11167  ℕcn 12145  ℤcz 12488  ℤ_≥cuz 12751  ..^cfzo 13570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-disj 5066  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571

This theorem is referenced by:  iunmbl  25510  volsup  25513  sigapildsys  34319  carsgclctunlem3  34477  voliunnfl  37865