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Theorem iundisjfi 28942
Description: Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 23123. (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 3650 . . . . . . 7 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (1..^𝑁)
2 fzossnn 12384 . . . . . . . . . 10 (1..^𝑁) ⊆ ℕ
3 nnuz 11599 . . . . . . . . . 10 ℕ = (ℤ‘1)
42, 3sseqtri 3600 . . . . . . . . 9 (1..^𝑁) ⊆ (ℤ‘1)
51, 4sstri 3577 . . . . . . . 8 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (ℤ‘1)
6 rabn0 3912 . . . . . . . . 9 ({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
76biimpri 217 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅)
8 infssuzcl 11648 . . . . . . . 8 (({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴})
95, 7, 8sylancr 694 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴})
101, 9sseldi 3566 . . . . . 6 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁))
11 nfrab1 3099 . . . . . . . . . . 11 𝑛{𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}
12 nfcv 2751 . . . . . . . . . . 11 𝑛
13 nfcv 2751 . . . . . . . . . . 11 𝑛 <
1411, 12, 13nfinf 8271 . . . . . . . . . 10 𝑛inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )
15 nfcv 2751 . . . . . . . . . 10 𝑛(1..^𝑁)
1614nfcsb1 3514 . . . . . . . . . . 11 𝑛inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
1716nfcri 2745 . . . . . . . . . 10 𝑛 𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
18 csbeq1a 3508 . . . . . . . . . . 11 (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝐴 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
1918eleq2d 2673 . . . . . . . . . 10 (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (𝑥𝐴𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
2014, 15, 17, 19elrabf 3329 . . . . . . . . 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 207 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
2221simprd 478 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
231, 2sstri 3577 . . . . . . . . . . 11 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ ℕ
24 nnssre 10901 . . . . . . . . . . 11 ℕ ⊆ ℝ
2523, 24sstri 3577 . . . . . . . . . 10 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ ℝ
2625, 9sseldi 3566 . . . . . . . . 9 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
2726ltnrd 10050 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ¬ inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
28 eliun 4460 . . . . . . . . 9 (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵)
2926ad2antrr 758 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
30 elfzouz2 12353 . . . . . . . . . . . . . . . . 17 (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) → 𝑁 ∈ (ℤ‘inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
31 fzoss2 12365 . . . . . . . . . . . . . . . . 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 3568 . . . . . . . . . . . . . . 15 ((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) → 𝑘 ∈ (1..^𝑁))
3433adantr 480 . . . . . . . . . . . . . 14 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ (1..^𝑁))
352, 34sseldi 3566 . . . . . . . . . . . . 13 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℕ)
3635nnred 10912 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℝ)
37 simpr 476 . . . . . . . . . . . . . 14 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑥𝐵)
38 nfcv 2751 . . . . . . . . . . . . . . 15 𝑛𝑘
39 iundisj3.0 . . . . . . . . . . . . . . . 16 𝑛𝐵
4039nfcri 2745 . . . . . . . . . . . . . . 15 𝑛 𝑥𝐵
41 iundisj3.1 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘𝐴 = 𝐵)
4241eleq2d 2673 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝑥𝐴𝑥𝐵))
4338, 15, 40, 42elrabf 3329 . . . . . . . . . . . . . 14 (𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ↔ (𝑘 ∈ (1..^𝑁) ∧ 𝑥𝐵))
4434, 37, 43sylanbrc 695 . . . . . . . . . . . . 13 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴})
45 infssuzle 11647 . . . . . . . . . . . . 13 (({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ 𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
465, 44, 45sylancr 694 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
47 elfzolt2 12348 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
4847ad2antlr 759 . . . . . . . . . . . 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 10074 . . . . . . . . . . 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 449 . . . . . . . . . 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 3013 . . . . . . . . 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 231 . . . . . . . 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 188 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ¬ 𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)
5422, 53eldifd 3551 . . . . . 6 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑥 ∈ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵))
55 csbeq1 3502 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝑚 / 𝑛𝐴 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
56 oveq2 6557 . . . . . . . . . 10 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
5756iuneq1d 4481 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝑘 ∈ (1..^𝑚)𝐵 = 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)
5855, 57difeq12d 3691 . . . . . . . 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 2673 . . . . . . 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 3282 . . . . . 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 691 . . . . 5 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
62 nfv 1830 . . . . . 6 𝑚 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
63 nfcsb1v 3515 . . . . . . . 8 𝑛𝑚 / 𝑛𝐴
64 nfcv 2751 . . . . . . . . 9 𝑛(1..^𝑚)
6564, 39nfiun 4484 . . . . . . . 8 𝑛 𝑘 ∈ (1..^𝑚)𝐵
6663, 65nfdif 3693 . . . . . . 7 𝑛(𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
6766nfcri 2745 . . . . . 6 𝑛 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
68 csbeq1a 3508 . . . . . . . 8 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
69 oveq2 6557 . . . . . . . . 9 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
7069iuneq1d 4481 . . . . . . . 8 (𝑛 = 𝑚 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑚)𝐵)
7168, 70difeq12d 3691 . . . . . . 7 (𝑛 = 𝑚 → (𝐴 𝑘 ∈ (1..^𝑛)𝐵) = (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
7271eleq2d 2673 . . . . . 6 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)))
7362, 67, 72cbvrex 3144 . . . . 5 (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
7461, 73sylibr 223 . . . 4 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
75 eldifi 3694 . . . . 5 (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥𝐴)
7675reximi 2994 . . . 4 (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
7774, 76impbii 198 . . 3 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
78 eliun 4460 . . 3 (𝑥 𝑛 ∈ (1..^𝑁)𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
79 eliun 4460 . . 3 (𝑥 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
8077, 78, 793bitr4i 291 . 2 (𝑥 𝑛 ∈ (1..^𝑁)𝐴𝑥 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵))
8180eqriv 2607 1 𝑛 ∈ (1..^𝑁)𝐴 = 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wnfc 2738  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  cr 9814  1c1 9816   < clt 9953  cle 9954  cn 10897  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:  iundisjcnt  28944
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