Theorem iundisj2 25526

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 1546	. . . 4 ⊢ ⊤
2		eqeq12 2754	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
3		csbeq1 3841	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
4		csbeq1 3841	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	3, 4	ineqan12d 4163	. . . . . . 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 2739	. . . . . 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 919	. . . . 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 2754	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑦 = 𝑥))
9		equcom 2020	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
10	8, 9	bitrdi 287	. . . . . 6 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
11		csbeq1 3841	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
12		csbeq1 3841	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
13	11, 12	ineqan12d 4163	. . . . . . . 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 4150	. . . . . . . 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 2788	. . . . . . 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 2739	. . . . . 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 919	. . . . 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 12169	. . . . . 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 2989	. . . . . . . 8 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 = 𝑦)
22		nnre 12172	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
23		nnre 12172	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
24		id 22	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑦)
25		leltne 11226	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
26	22, 23, 24, 25	syl3an 1161	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
27		vex 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
28		nfcsb1v 3862	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑥 / 𝑛⦌𝐴
29		nfcv 2899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑥)𝐵
30	28, 29	nfdif 4070	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
31		csbeq1a 3852	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → 𝐴 = ⦋𝑥 / 𝑛⦌𝐴)
32		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑥 → (1..^𝑛) = (1..^𝑥))
33	32	iuneq1d 4962	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑥)𝐵)
34	31, 33	difeq12d 4068	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵))
35	27, 30, 34	csbief 3872	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
36		vex 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
37		nfcsb1v 3862	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴
38		nfcv 2899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑦)𝐵
39	37, 38	nfdif 4070	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
40		csbeq1a 3852	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴)
41		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → (1..^𝑛) = (1..^𝑦))
42	41	iuneq1d 4962	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑦)𝐵)
43	40, 42	difeq12d 4068	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵))
44	36, 39, 43	csbief 3872	. . . . . . . . . . . . . 14 ⊢ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
45	35, 44	ineq12i 4159	. . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℕ)
47		nnuz 12818	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ = (ℤ≥‘1)
48	46, 47	eleqtrdi 2847	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (ℤ≥‘1))
49		simp2 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℕ)
50	49	nnzd 12541	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℤ)
51		simp3 1139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
52		elfzo2 13607	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1..^𝑦) ↔ (𝑥 ∈ (ℤ≥‘1) ∧ 𝑦 ∈ ℤ ∧ 𝑥 < 𝑦))
53	48, 50, 51, 52	syl3anbrc 1345	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1..^𝑦))
54		nfcv 2899	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝑘
55		nfcv 2899	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝐵
56		iundisj.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)
57	54, 55, 56	csbhypf 3866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
58	57	equcoms 2022	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
59	58	eqcomd 2743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑛⦌𝐴)
60	59	ssiun2s 4992	. . . . . . . . . . . . . . . 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 4088	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵) ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
63	62	ssrind 4185	. . . . . . . . . . . . 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 3961	. . . . . . . . . . . 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 4413	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) = ∅
66		sseq0 4344	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
68	67	3expia 1122	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
69	68	3adant3 1133	. . . . . . . . 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 864	. . . . . 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 11674	. . . 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 691	. . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
76	75	rgen2 3178	. 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
77		disjors 5069	. 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 848   ∧ w3a 1087   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ⦋csb 3838   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ∪ ciun 4934  Disj wdisj 5053   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  ℝcr 11028  1c1 11030   < clt 11170   ≤ cle 11171  ℕcn 12165  ℤcz 12515  ℤ_≥cuz 12779  ..^cfzo 13599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600

This theorem is referenced by:  iunmbl  25530  volsup  25533  sigapildsys  34322  carsgclctunlem3  34480  voliunnfl  37999