iundisjfi - Mathbox for Thierry Arnoux

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Theorem iundisjfi 32514

Description: Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 25432. (Contributed by Thierry Arnoux, 15-Feb-2017.)

**Hypotheses**
Ref	Expression
iundisj3.0	⊢ Ⅎ𝑛𝐵
iundisj3.1	⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)

**Assertion**
Ref	Expression
iundisjfi	⊢ ∪ 𝑛 ∈ (1..^𝑁)𝐴 = ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Distinct variable groups: 𝑘,𝑛,𝑁 𝐴,𝑘 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑘,𝑛)

Proof of Theorem iundisjfi Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4072	. . . . . . 7 ⊢ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ⊆ (1..^𝑁)
2		fzossnn 13687	. . . . . . . . . 10 ⊢ (1..^𝑁) ⊆ ℕ
3		nnuz 12869	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
4	2, 3	sseqtri 4013	. . . . . . . . 9 ⊢ (1..^𝑁) ⊆ (ℤ_≥‘1)
5	1, 4	sstri 3986	. . . . . . . 8 ⊢ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ⊆ (ℤ_≥‘1)
6		rabn0 4380	. . . . . . . . 9 ⊢ ({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ≠ ∅ ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴)
7	6	biimpri 227	. . . . . . . 8 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ≠ ∅)
8		infssuzcl 12920	. . . . . . . 8 ⊢ (({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ⊆ (ℤ_≥‘1) ∧ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ≠ ∅) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴})
9	5, 7, 8	sylancr 586	. . . . . . 7 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴})
10	1, 9	sselid 3975	. . . . . 6 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ (1..^𝑁))
11		nfrab1 3445	. . . . . . . . . . 11 ⊢ Ⅎ𝑛{𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}
12		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ℝ
13		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 <
14	11, 12, 13	nfinf 9479	. . . . . . . . . 10 ⊢ Ⅎ𝑛inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )
15		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑛(1..^𝑁)
16	14	nfcsb1 3912	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴
17	16	nfcri 2884	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑥 ∈ ⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴
18		csbeq1a 3902	. . . . . . . . . . 11 ⊢ (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → 𝐴 = ⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴)
19	18	eleq2d 2813	. . . . . . . . . 10 ⊢ (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴))
20	14, 15, 17, 19	elrabf 3674	. . . . . . . . 9 ⊢ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ↔ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥 ∈ ⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴))
21	9, 20	sylib 217	. . . . . . . 8 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥 ∈ ⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴))
22	21	simprd 495	. . . . . . 7 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → 𝑥 ∈ ⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴)
23	1, 2	sstri 3986	. . . . . . . . . . 11 ⊢ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ⊆ ℕ
24		nnssre 12220	. . . . . . . . . . 11 ⊢ ℕ ⊆ ℝ
25	23, 24	sstri 3986	. . . . . . . . . 10 ⊢ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ⊆ ℝ
26	25, 9	sselid 3975	. . . . . . . . 9 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
27	26	ltnrd 11352	. . . . . . . 8 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → ¬ inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
28		eliun 4994	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝑥 ∈ 𝐵)
29	26	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
30		elfzouz2 13653	. . . . . . . . . . . . . . . 16 ⊢ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ (1..^𝑁) → 𝑁 ∈ (ℤ_≥‘inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
31		fzoss2 13666	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )) → (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )) ⊆ (1..^𝑁))
32	10, 30, 31	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )) ⊆ (1..^𝑁))
33	32	sselda 3977	. . . . . . . . . . . . . 14 ⊢ ((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) → 𝑘 ∈ (1..^𝑁))
34	33	adantr 480	. . . . . . . . . . . . 13 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ (1..^𝑁))
35	2, 34	sselid 3975	. . . . . . . . . . . 12 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℕ)
36	35	nnred 12231	. . . . . . . . . . 11 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℝ)
37		simpr 484	. . . . . . . . . . . . 13 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
38		nfcv 2897	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑘
39		iundisj3.0	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝐵
40	39	nfcri 2884	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 𝑥 ∈ 𝐵
41		iundisj3.1	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)
42	41	eleq2d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
43	38, 15, 40, 42	elrabf 3674	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ↔ (𝑘 ∈ (1..^𝑁) ∧ 𝑥 ∈ 𝐵))
44	34, 37, 43	sylanbrc 582	. . . . . . . . . . . 12 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴})
45		infssuzle 12919	. . . . . . . . . . . 12 ⊢ (({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴} ⊆ (ℤ_≥‘1) ∧ 𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ≤ 𝑘)
46	5, 44, 45	sylancr 586	. . . . . . . . . . 11 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ≤ 𝑘)
47		elfzolt2 13647	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
48	47	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
49	29, 36, 29, 46, 48	lelttrd 11376	. . . . . . . . . 10 ⊢ (((∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
50	49	rexlimdva2 3151	. . . . . . . . 9 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → (∃𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝑥 ∈ 𝐵 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
51	28, 50	biimtrid 241	. . . . . . . 8 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → (𝑥 ∈ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
52	27, 51	mtod 197	. . . . . . 7 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)
53	22, 52	eldifd 3954	. . . . . 6 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → 𝑥 ∈ (⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵))
54		csbeq1 3891	. . . . . . . . 9 ⊢ (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → ⦋𝑚 / 𝑛⦌𝐴 = ⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴)
55		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
56	55	iuneq1d 5017	. . . . . . . . 9 ⊢ (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → ∪ 𝑘 ∈ (1..^𝑚)𝐵 = ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)
57	54, 56	difeq12d 4118	. . . . . . . 8 ⊢ (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵) = (⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵))
58	57	eleq2d 2813	. . . . . . 7 ⊢ (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵) ↔ 𝑥 ∈ (⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)))
59	58	rspcev 3606	. . . . . 6 ⊢ ((inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥 ∈ (⦋inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)) → ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
60	10, 53, 59	syl2anc 583	. . . . 5 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
61		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑚 𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
62		nfcsb1v 3913	. . . . . . . 8 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
63		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑛(1..^𝑚)
64	63, 39	nfiun 5020	. . . . . . . 8 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑚)𝐵
65	62, 64	nfdif 4120	. . . . . . 7 ⊢ Ⅎ𝑛(⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵)
66	65	nfcri 2884	. . . . . 6 ⊢ Ⅎ𝑛 𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵)
67		csbeq1a 3902	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
68		oveq2 7413	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
69	68	iuneq1d 5017	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑚)𝐵)
70	67, 69	difeq12d 4118	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
71	70	eleq2d 2813	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵)))
72	61, 66, 71	cbvrexw 3298	. . . . 5 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
73	60, 72	sylibr 233	. . . 4 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 → ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
74		eldifi 4121	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥 ∈ 𝐴)
75	74	reximi 3078	. . . 4 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) → ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴)
76	73, 75	impbii 208	. . 3 ⊢ (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
77		eliun 4994	. . 3 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (1..^𝑁)𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ 𝐴)
78		eliun 4994	. . 3 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
79	76, 77, 78	3bitr4i 303	. 2 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (1..^𝑁)𝐴 ↔ 𝑥 ∈ ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
80	79	eqriv 2723	1 ⊢ ∪ 𝑛 ∈ (1..^𝑁)𝐴 = ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2877 ≠ wne 2934 ∃wrex 3064 {crab 3426 ⦋csb 3888 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 ∪ ciun 4990 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 infcinf 9438 ℝcr 11111 1c1 11113 < clt 11252 ≤ cle 11253 ℕcn 12216 ℤ_≥cuz 12826 ..^cfzo 13633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634

This theorem is referenced by: iundisjcnt 32516

W3C validator