Theorem iundisj2f 30929

Description: A disjoint union is disjoint. Cf. iundisj2 24713. (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 1543	. . . 4 ⊢ ⊤
2		eqeq12 2755	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
3		csbeq1 3835	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
4		csbeq1 3835	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	3, 4	ineqan12d 4148	. . . . . . 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 2740	. . . . . 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 916	. . . . 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 2755	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑦 = 𝑥))
9		equcom 2021	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
10	8, 9	bitrdi 287	. . . . . 6 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
11		csbeq1 3835	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
12		csbeq1 3835	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
13	11, 12	ineqan12d 4148	. . . . . . . 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 4135	. . . . . . . 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 2794	. . . . . . 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 2740	. . . . . 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 916	. . . . 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 11977	. . . . . 6 ⊢ ℕ ⊆ ℝ
19	18	a1i 11	. . . . 5 ⊢ (⊤ → ℕ ⊆ ℝ)
20		biidd 261	. . . . 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 3000	. . . . . . . 8 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 = 𝑦)
22		nnre 11980	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
23		nnre 11980	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
24		id 22	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑦)
25		leltne 11064	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
26	22, 23, 24, 25	syl3an 1159	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
27		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
28		nfcsb1v 3857	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑥 / 𝑛⦌𝐴
29		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(1..^𝑥)
30		iundisjf.2	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛𝐵
31	29, 30	nfiun 4954	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑥)𝐵
32	28, 31	nfdif 4060	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
33		csbeq1a 3846	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → 𝐴 = ⦋𝑥 / 𝑛⦌𝐴)
34		oveq2 7283	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑥 → (1..^𝑛) = (1..^𝑥))
35	34	iuneq1d 4951	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑥)𝐵)
36	33, 35	difeq12d 4058	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵))
37	27, 32, 36	csbief 3867	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
38		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
39		nfcsb1v 3857	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴
40		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(1..^𝑦)
41	40, 30	nfiun 4954	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑦)𝐵
42	39, 41	nfdif 4060	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
43		csbeq1a 3846	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴)
44		oveq2 7283	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → (1..^𝑛) = (1..^𝑦))
45	44	iuneq1d 4951	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑦)𝐵)
46	43, 45	difeq12d 4058	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵))
47	38, 42, 46	csbief 3867	. . . . . . . . . . . . . 14 ⊢ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
48	37, 47	ineq12i 4144	. . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℕ)
50		nnuz 12621	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ = (ℤ≥‘1)
51	49, 50	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (ℤ≥‘1))
52		simp2 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℕ)
53	52	nnzd 12425	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℤ)
54		simp3 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
55		elfzo2 13390	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1..^𝑦) ↔ (𝑥 ∈ (ℤ≥‘1) ∧ 𝑦 ∈ ℤ ∧ 𝑥 < 𝑦))
56	51, 53, 54, 55	syl3anbrc 1342	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1..^𝑦))
57		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(1..^𝑦)
58		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑥
59		iundisjf.1	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘𝐴
60	58, 59	nfcsbw 3859	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑥 / 𝑛⦌𝐴
61		nfcv 2907	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝑘
62		iundisjf.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)
63	61, 30, 62	csbhypf 3861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
64	63	equcoms 2023	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
65	64	eqcomd 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑛⦌𝐴)
66	57, 58, 60, 65	ssiun2sf 30899	. . . . . . . . . . . . . . . 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 4077	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵) ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
69	68	ssrind 4169	. . . . . . . . . . . . 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 3969	. . . . . . . . . . . 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 4405	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) = ∅
72		sseq0 4333	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
74	73	3expia 1120	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
75	74	3adant3 1131	. . . . . . . . 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 259	. . . . . . . 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	syl5bir 242	. . . . . . 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 860	. . . . . 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 11508	. . . 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 687	. . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
82	81	rgen2 3120	. 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
83		disjors 5055	. 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 230	1 ⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  Ⅎwnfc 2887   ≠ wne 2943  ∀wral 3064  ⦋csb 3832   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ∪ ciun 4924  Disj wdisj 5039   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  ℝcr 10870  1c1 10872   < clt 11009   ≤ cle 11010  ℕcn 11973  ℤcz 12319  ℤ_≥cuz 12582  ..^cfzo 13382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383

This theorem is referenced by:  iundisj2cnt  31120