Theorem iundisj2f 32686

Description: A disjoint union is disjoint. Cf. iundisj2 25541. (Contributed by Thierry Arnoux, 30-Dec-2016.)

Hypotheses

Ref	Expression
iundisjf.1	⊢ Ⅎ𝑘𝐴
iundisjf.2	⊢ Ⅎ𝑛𝐵
iundisjf.3	⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)

Assertion

Ref	Expression
iundisj2f	⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Distinct variable group:   𝑘,𝑛

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

Proof of Theorem iundisj2f

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

Step	Hyp	Ref	Expression
1		tru 1551	. . . 4 ⊢ ⊤
2		eqeq12 2757	. . . . . 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 4158	. . . . . . 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 2742	. . . . . 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 924	. . . . 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 2757	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑦 = 𝑥))
9		equcom 2025	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
10	8, 9	bitrdi 288	. . . . . 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 4158	. . . . . . . 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 4145	. . . . . . . 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 2791	. . . . . . 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 2742	. . . . . 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 924	. . . . 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 12176	. . . . . 6 ⊢ ℕ ⊆ ℝ
19	18	a1i 11	. . . . 5 ⊢ (⊤ → ℕ ⊆ ℝ)
20		biidd 263	. . . . 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 2991	. . . . . . . 8 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 = 𝑦)
22		nnre 12179	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
23		nnre 12179	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
24		id 22	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑦)
25		leltne 11233	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
26	22, 23, 24, 25	syl3an 1166	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
27		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
28		nfcsb1v 3862	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑥 / 𝑛⦌𝐴
29		nfcv 2902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(1..^𝑥)
30		iundisjf.2	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛𝐵
31	29, 30	nfiun 4960	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑥)𝐵
32	28, 31	nfdif 4067	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
33		csbeq1a 3852	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → 𝐴 = ⦋𝑥 / 𝑛⦌𝐴)
34		oveq2 7371	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑥 → (1..^𝑛) = (1..^𝑥))
35	34	iuneq1d 4956	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑥)𝐵)
36	33, 35	difeq12d 4065	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵))
37	27, 32, 36	csbief 3872	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
38		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
39		nfcsb1v 3862	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴
40		nfcv 2902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(1..^𝑦)
41	40, 30	nfiun 4960	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑦)𝐵
42	39, 41	nfdif 4067	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
43		csbeq1a 3852	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴)
44		oveq2 7371	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → (1..^𝑛) = (1..^𝑦))
45	44	iuneq1d 4956	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑦)𝐵)
46	43, 45	difeq12d 4065	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵))
47	38, 42, 46	csbief 3872	. . . . . . . . . . . . . 14 ⊢ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
48	37, 47	ineq12i 4154	. . . . . . . . . . . . 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))
49		simp1 1142	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℕ)
50		nnuz 12825	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ = (ℤ≥‘1)
51	49, 50	eleqtrdi 2850	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (ℤ≥‘1))
52		simp2 1143	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℕ)
53	52	nnzd 12548	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℤ)
54		simp3 1144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
55		elfzo2 13614	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1..^𝑦) ↔ (𝑥 ∈ (ℤ≥‘1) ∧ 𝑦 ∈ ℤ ∧ 𝑥 < 𝑦))
56	51, 53, 54, 55	syl3anbrc 1350	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1..^𝑦))
57		nfcv 2902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(1..^𝑦)
58		nfcv 2902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑥
59		iundisjf.1	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘𝐴
60	58, 59	nfcsbw 3864	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑥 / 𝑛⦌𝐴
61		nfcv 2902	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝑘
62		iundisjf.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)
63	61, 30, 62	csbhypf 3866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
64	63	equcoms 2027	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
65	64	eqcomd 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑛⦌𝐴)
66	57, 58, 60, 65	ssiun2sf 32655	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1..^𝑦) → ⦋𝑥 / 𝑛⦌𝐴 ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
67	56, 66	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → ⦋𝑥 / 𝑛⦌𝐴 ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
68	67	ssdifssd 4084	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵) ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
69	68	ssrind 4179	. . . . . . . . . . . . 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)))
70	48, 69	eqsstrid 3960	. . . . . . . . . . . 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)))
71		disjdif 4407	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) = ∅
72		sseq0 4338	. . . . . . . . . . . 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)) = ∅)
73	70, 71, 72	sylancl 592	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
74	73	3expia 1127	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
75	74	3adant3 1138	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
76	26, 75	sylbird 261	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑦 ≠ 𝑥 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
77	21, 76	biimtrrid 244	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (¬ 𝑥 = 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
78	77	orrd 869	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
79	78	adantl 482	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦)) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
80	7, 17, 19, 20, 79	wlogle 11681	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
81	1, 80	mpan 696	. . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
82	81	rgen2 3180	. 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
83		disjors 5062	. 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)) = ∅))
84	82, 83	mpbir 232	1 ⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  Ⅎwnfc 2887   ≠ wne 2935  ∀wral 3054  ⦋csb 3838   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  ∪ ciun 4928  Disj wdisj 5046   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  ℝcr 11035  1c1 11037   < clt 11177   ≤ cle 11178  ℕcn 12172  ℤcz 12522  ℤ_≥cuz 12786  ..^cfzo 13606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-disj 5047  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607

This theorem is referenced by:  iundisj2cnt  32898