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Theorem iundisjfi 28767
Description: Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 23026. (Contributed by Thierry Arnoux, 15-Feb-2017.)
Hypotheses
Ref Expression
iundisj3.0 𝑛𝐵
iundisj3.1 (𝑛 = 𝑘𝐴 = 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.
StepHypRef Expression
1 ssrab2 3554 . . . . . . 7 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (1..^𝑁)
2 fzossnn 12249 . . . . . . . . . 10 (1..^𝑁) ⊆ ℕ
3 nnuz 11459 . . . . . . . . . 10 ℕ = (ℤ‘1)
42, 3sseqtri 3504 . . . . . . . . 9 (1..^𝑁) ⊆ (ℤ‘1)
51, 4sstri 3481 . . . . . . . 8 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (ℤ‘1)
6 rabn0 3815 . . . . . . . . 9 ({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
76biimpri 216 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅)
8 infssuzcl 11508 . . . . . . . 8 (({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴})
95, 7, 8sylancr 693 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴})
101, 9sseldi 3470 . . . . . 6 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁))
11 nfrab1 3003 . . . . . . . . . . 11 𝑛{𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}
12 nfcv 2655 . . . . . . . . . . 11 𝑛
13 nfcv 2655 . . . . . . . . . . 11 𝑛 <
1411, 12, 13nfinf 8145 . . . . . . . . . 10 𝑛inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )
15 nfcv 2655 . . . . . . . . . 10 𝑛(1..^𝑁)
1614nfcsb1 3418 . . . . . . . . . . 11 𝑛inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
1716nfcri 2649 . . . . . . . . . 10 𝑛 𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
18 csbeq1a 3412 . . . . . . . . . . 11 (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝐴 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
1918eleq2d 2577 . . . . . . . . . 10 (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (𝑥𝐴𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
2014, 15, 17, 19elrabf 3233 . . . . . . . . 9 (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ↔ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
219, 20sylib 206 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
2221simprd 477 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
231, 2sstri 3481 . . . . . . . . . . 11 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ ℕ
24 nnssre 10777 . . . . . . . . . . 11 ℕ ⊆ ℝ
2523, 24sstri 3481 . . . . . . . . . 10 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ ℝ
2625, 9sseldi 3470 . . . . . . . . 9 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
2726ltnrd 9920 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ¬ inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
28 eliun 4358 . . . . . . . . 9 (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵)
2926ad2antrr 757 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
30 elfzouz2 12218 . . . . . . . . . . . . . . . . 17 (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) → 𝑁 ∈ (ℤ‘inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
31 fzoss2 12230 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )) → (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )) ⊆ (1..^𝑁))
3210, 30, 313syl 18 . . . . . . . . . . . . . . . 16 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )) ⊆ (1..^𝑁))
3332sselda 3472 . . . . . . . . . . . . . . 15 ((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) → 𝑘 ∈ (1..^𝑁))
3433adantr 479 . . . . . . . . . . . . . 14 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ (1..^𝑁))
352, 34sseldi 3470 . . . . . . . . . . . . 13 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℕ)
3635nnred 10788 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℝ)
37 simpr 475 . . . . . . . . . . . . . 14 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑥𝐵)
38 nfcv 2655 . . . . . . . . . . . . . . 15 𝑛𝑘
39 iundisj3.0 . . . . . . . . . . . . . . . 16 𝑛𝐵
4039nfcri 2649 . . . . . . . . . . . . . . 15 𝑛 𝑥𝐵
41 iundisj3.1 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘𝐴 = 𝐵)
4241eleq2d 2577 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝑥𝐴𝑥𝐵))
4338, 15, 40, 42elrabf 3233 . . . . . . . . . . . . . 14 (𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ↔ (𝑘 ∈ (1..^𝑁) ∧ 𝑥𝐵))
4434, 37, 43sylanbrc 694 . . . . . . . . . . . . 13 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴})
45 infssuzle 11507 . . . . . . . . . . . . 13 (({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ 𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
465, 44, 45sylancr 693 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
47 elfzolt2 12213 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
4847ad2antlr 758 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
4929, 36, 29, 46, 48lelttrd 9944 . . . . . . . . . . 11 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
5049ex 448 . . . . . . . . . 10 ((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) → (𝑥𝐵 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
5150rexlimdva 2917 . . . . . . . . 9 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → (∃𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
5228, 51syl5bi 230 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
5327, 52mtod 187 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ¬ 𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)
5422, 53eldifd 3455 . . . . . 6 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑥 ∈ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵))
55 csbeq1 3406 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝑚 / 𝑛𝐴 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
56 oveq2 6433 . . . . . . . . . 10 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
5756iuneq1d 4379 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝑘 ∈ (1..^𝑚)𝐵 = 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)
5855, 57difeq12d 3595 . . . . . . . 8 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵) = (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵))
5958eleq2d 2577 . . . . . . 7 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵) ↔ 𝑥 ∈ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)))
6059rspcev 3186 . . . . . 6 ((inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥 ∈ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)) → ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
6110, 54, 60syl2anc 690 . . . . 5 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
62 nfv 1796 . . . . . 6 𝑚 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
63 nfcsb1v 3419 . . . . . . . 8 𝑛𝑚 / 𝑛𝐴
64 nfcv 2655 . . . . . . . . 9 𝑛(1..^𝑚)
6564, 39nfiun 4382 . . . . . . . 8 𝑛 𝑘 ∈ (1..^𝑚)𝐵
6663, 65nfdif 3597 . . . . . . 7 𝑛(𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
6766nfcri 2649 . . . . . 6 𝑛 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
68 csbeq1a 3412 . . . . . . . 8 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
69 oveq2 6433 . . . . . . . . 9 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
7069iuneq1d 4379 . . . . . . . 8 (𝑛 = 𝑚 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑚)𝐵)
7168, 70difeq12d 3595 . . . . . . 7 (𝑛 = 𝑚 → (𝐴 𝑘 ∈ (1..^𝑛)𝐵) = (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
7271eleq2d 2577 . . . . . 6 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)))
7362, 67, 72cbvrex 3048 . . . . 5 (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
7461, 73sylibr 222 . . . 4 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
75 eldifi 3598 . . . . 5 (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥𝐴)
7675reximi 2898 . . . 4 (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
7774, 76impbii 197 . . 3 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
78 eliun 4358 . . 3 (𝑥 𝑛 ∈ (1..^𝑁)𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
79 eliun 4358 . . 3 (𝑥 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
8077, 78, 793bitr4i 290 . 2 (𝑥 𝑛 ∈ (1..^𝑁)𝐴𝑥 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵))
8180eqriv 2511 1 𝑛 ∈ (1..^𝑁)𝐴 = 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  wnfc 2642  wne 2684  wrex 2801  {crab 2804  csb 3403  cdif 3441  wss 3444  c0 3777   ciun 4353   class class class wbr 4481  cfv 5689  (class class class)co 6425  infcinf 8104  cr 9688  1c1 9690   < clt 9827  cle 9828  cn 10773  cuz 11423  ..^cfzo 12199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6832  df-1st 6932  df-2nd 6933  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-er 7503  df-en 7716  df-dom 7717  df-sdom 7718  df-sup 8105  df-inf 8106  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-nn 10774  df-n0 11046  df-z 11117  df-uz 11424  df-fz 12063  df-fzo 12200
This theorem is referenced by:  iundisjcnt  28769
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