Theorem iundisj2 23833

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 1526	. . . 4 ⊢ ⊤
2		eqeq12 2808	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
3		csbeq1 3814	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
4		csbeq1 3814	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	3, 4	ineqan12d 4111	. . . . . . 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 2797	. . . . . 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 913	. . . . 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 2808	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑦 = 𝑥))
9		equcom 2002	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
10	8, 9	syl6bb 288	. . . . . 6 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑎 = 𝑏 ↔ 𝑥 = 𝑦))
11		csbeq1 3814	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ⦋𝑎 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
12		csbeq1 3814	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → ⦋𝑏 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
13	11, 12	ineqan12d 4111	. . . . . . . 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 4099	. . . . . . . 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	syl6eq 2847	. . . . . . 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 2797	. . . . . 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 913	. . . . 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 11490	. . . . . 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 3040	. . . . . . . 8 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑥 = 𝑦)
22		nnre 11493	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
23		nnre 11493	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
24		id 22	. . . . . . . . . 10 ⊢ (𝑥 ≤ 𝑦 → 𝑥 ≤ 𝑦)
25		leltne 10577	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
26	22, 23, 24, 25	syl3an 1153	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 ≤ 𝑦) → (𝑥 < 𝑦 ↔ 𝑦 ≠ 𝑥))
27		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
28		nfcsb1v 3833	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑥 / 𝑛⦌𝐴
29		nfcv 2949	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑥)𝐵
30	28, 29	nfdif 4023	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
31		csbeq1a 3824	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → 𝐴 = ⦋𝑥 / 𝑛⦌𝐴)
32		oveq2 7024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑥 → (1..^𝑛) = (1..^𝑥))
33	32	iuneq1d 4851	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑥 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑥)𝐵)
34	31, 33	difeq12d 4021	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑥 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵))
35	27, 30, 34	csbief 3842	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵)
36		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
37		nfcsb1v 3833	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑦 / 𝑛⦌𝐴
38		nfcv 2949	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑦)𝐵
39	37, 38	nfdif 4023	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
40		csbeq1a 3824	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑛⦌𝐴)
41		oveq2 7024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → (1..^𝑛) = (1..^𝑦))
42	41	iuneq1d 4851	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑦)𝐵)
43	40, 42	difeq12d 4021	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵))
44	36, 39, 43	csbief 3842	. . . . . . . . . . . . . 14 ⊢ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
45	35, 44	ineq12i 4107	. . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℕ)
47		nnuz 12130	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ = (ℤ≥‘1)
48	46, 47	syl6eleq 2893	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (ℤ≥‘1))
49		simp2 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℕ)
50	49	nnzd 11935	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℤ)
51		simp3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
52		elfzo2 12891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1..^𝑦) ↔ (𝑥 ∈ (ℤ≥‘1) ∧ 𝑦 ∈ ℤ ∧ 𝑥 < 𝑦))
53	48, 50, 51, 52	syl3anbrc 1336	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1..^𝑦))
54		nfcv 2949	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝑘
55		nfcv 2949	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑛𝐵
56		iundisj.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)
57	54, 55, 56	csbhypf 3836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
58	57	equcoms 2004	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → ⦋𝑥 / 𝑛⦌𝐴 = 𝐵)
59	58	eqcomd 2801	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑛⦌𝐴)
60	59	ssiun2s 4871	. . . . . . . . . . . . . . . 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 4040	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑥 < 𝑦) → (⦋𝑥 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑥)𝐵) ⊆ ∪ 𝑘 ∈ (1..^𝑦)𝐵)
63	62	ssrind 4132	. . . . . . . . . . . . 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	syl5eqss 3936	. . . . . . . . . . . 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 4335	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ (1..^𝑦)𝐵 ∩ (⦋𝑦 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑦)𝐵)) = ∅
66		sseq0 4273	. . . . . . . . . . . 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 1114	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 → (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
69	68	3adant3 1125	. . . . . . . . 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 261	. . . . . . . 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	syl5bir 244	. . . . . . 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 858	. . . . . 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 482	. . . . 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 11021	. . . 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 686	. . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅))
76	75	rgen2a 3193	. 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 = 𝑦 ∨ (⦋𝑥 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∩ ⦋𝑦 / 𝑛⦌(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = ∅)
77		disjors 4945	. 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 232	1 ⊢ Disj 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080   = wceq 1522  ⊤wtru 1523   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  ⦋csb 3811   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  ∅c0 4211  ∪ ciun 4825  Disj wdisj 4930   class class class wbr 4962  ‘cfv 6225  (class class class)co 7016  ℝcr 10382  1c1 10384   < clt 10521   ≤ cle 10522  ℕcn 11486  ℤcz 11829  ℤ_≥cuz 12093  ..^cfzo 12883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-disj 4931  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-n0 11746  df-z 11830  df-uz 12094  df-fz 12743  df-fzo 12884

This theorem is referenced by:  iunmbl  23837  volsup  23840  sigapildsys  31038  carsgclctunlem3  31195  voliunnfl  34467