Theorem iundisj 23123

Description: Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)

Hypothesis

Ref	Expression
iundisj.1	⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)

Assertion

Ref	Expression
iundisj	⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 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 iundisj

Dummy variables 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3650	. . . . . . . . . 10 ⊢ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ⊆ ℕ
2		nnuz 11599	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
3	1, 2	sseqtri 3600	. . . . . . . . 9 ⊢ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ⊆ (ℤ≥‘1)
4		rabn0 3912	. . . . . . . . . 10 ⊢ ({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ≠ ∅ ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴)
5	4	biimpri 217	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ≠ ∅)
6		infssuzcl 11648	. . . . . . . . 9 ⊢ (({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ⊆ (ℤ≥‘1) ∧ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴})
7	3, 5, 6	sylancr 694	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴})
8		nfrab1 3099	. . . . . . . . . 10 ⊢ Ⅎ𝑛{𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}
9		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑛ℝ
10		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑛 <
11	8, 9, 10	nfinf 8271	. . . . . . . . 9 ⊢ Ⅎ𝑛inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )
12		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑛ℕ
13	11	nfcsb1 3514	. . . . . . . . . 10 ⊢ Ⅎ𝑛⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴
14	13	nfcri 2745	. . . . . . . . 9 ⊢ Ⅎ𝑛 𝑥 ∈ ⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴
15		csbeq1a 3508	. . . . . . . . . 10 ⊢ (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → 𝐴 = ⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴)
16	15	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴))
17	11, 12, 14, 16	elrabf 3329	. . . . . . . 8 ⊢ (inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ↔ (inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥 ∈ ⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴))
18	7, 17	sylib 207	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → (inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥 ∈ ⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴))
19	18	simpld 474	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℕ)
20	18	simprd 478	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → 𝑥 ∈ ⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴)
21	19	nnred 10912	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
22	21	ltnrd 10050	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → ¬ inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
23		eliun 4460	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝑥 ∈ 𝐵)
24	21	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
25		elfzouz 12343	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )) → 𝑘 ∈ (ℤ≥‘1))
26	25, 2	syl6eleqr 2699	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )) → 𝑘 ∈ ℕ)
27	26	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℕ)
28	27	nnred 10912	. . . . . . . . . . . 12 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℝ)
29		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
30		iundisj.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)
31	30	eleq2d 2673	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
32	31	elrab 3331	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ↔ (𝑘 ∈ ℕ ∧ 𝑥 ∈ 𝐵))
33	27, 29, 32	sylanbrc 695	. . . . . . . . . . . . 13 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴})
34		infssuzle 11647	. . . . . . . . . . . . 13 ⊢ (({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴} ⊆ (ℤ≥‘1) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}) → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ≤ 𝑘)
35	3, 33, 34	sylancr 694	. . . . . . . . . . . 12 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ≤ 𝑘)
36		elfzolt2 12348	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
37	36	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
38	24, 28, 24, 35, 37	lelttrd 10074	. . . . . . . . . . 11 ⊢ (((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) ∧ 𝑥 ∈ 𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))
39	38	ex 449	. . . . . . . . . 10 ⊢ ((∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))) → (𝑥 ∈ 𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
40	39	rexlimdva 3013	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → (∃𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝑥 ∈ 𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
41	23, 40	syl5bi 231	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → (𝑥 ∈ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
42	22, 41	mtod 188	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)
43	20, 42	eldifd 3551	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → 𝑥 ∈ (⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵))
44		csbeq1 3502	. . . . . . . . 9 ⊢ (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → ⦋𝑚 / 𝑛⦌𝐴 = ⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴)
45		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < )))
46	45	iuneq1d 4481	. . . . . . . . 9 ⊢ (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → ∪ 𝑘 ∈ (1..^𝑚)𝐵 = ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)
47	44, 46	difeq12d 3691	. . . . . . . 8 ⊢ (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵) = (⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵))
48	47	eleq2d 2673	. . . . . . 7 ⊢ (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) → (𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵) ↔ 𝑥 ∈ (⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)))
49	48	rspcev 3282	. . . . . 6 ⊢ ((inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥 ∈ (⦋inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ) / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥 ∈ 𝐴}, ℝ, < ))𝐵)) → ∃𝑚 ∈ ℕ 𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
50	19, 43, 49	syl2anc 691	. . . . 5 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
51		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑚 𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
52		nfcsb1v 3515	. . . . . . . 8 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
53		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑛∪ 𝑘 ∈ (1..^𝑚)𝐵
54	52, 53	nfdif 3693	. . . . . . 7 ⊢ Ⅎ𝑛(⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵)
55	54	nfcri 2745	. . . . . 6 ⊢ Ⅎ𝑛 𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵)
56		csbeq1a 3508	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
57		oveq2 6557	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
58	57	iuneq1d 4481	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ∪ 𝑘 ∈ (1..^𝑛)𝐵 = ∪ 𝑘 ∈ (1..^𝑚)𝐵)
59	56, 58	difeq12d 3691	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) = (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
60	59	eleq2d 2673	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵)))
61	51, 55, 60	cbvrex 3144	. . . . 5 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (⦋𝑚 / 𝑛⦌𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑚)𝐵))
62	50, 61	sylibr 223	. . . 4 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 → ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
63		eldifi 3694	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥 ∈ 𝐴)
64	63	reximi 2994	. . . 4 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) → ∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴)
65	62, 64	impbii 198	. . 3 ⊢ (∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
66		eliun 4460	. . 3 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ 𝐴)
67		eliun 4460	. . 3 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
68	65, 66, 67	3bitr4i 291	. 2 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ 𝐴 ↔ 𝑥 ∈ ∪ 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
69	68	eqriv 2607	1 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑛 ∈ ℕ (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  ⦋csb 3499   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  infcinf 8230  ℝcr 9814  1c1 9816   < clt 9953   ≤ cle 9954  ℕcn 10897  ℤ_≥cuz 11563  ..^cfzo 12334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335

This theorem is referenced by:  iunmbl  23128  volsup  23131  sigapildsys  29552  carsgclctunlem3  29709  voliunnfl  32623