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Theorem iundisjfi 29529
Description: Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 23297. (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 3679 . . . . . . 7 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (1..^𝑁)
2 fzossnn 12500 . . . . . . . . . 10 (1..^𝑁) ⊆ ℕ
3 nnuz 11708 . . . . . . . . . 10 ℕ = (ℤ‘1)
42, 3sseqtri 3629 . . . . . . . . 9 (1..^𝑁) ⊆ (ℤ‘1)
51, 4sstri 3604 . . . . . . . 8 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ (ℤ‘1)
6 rabn0 3949 . . . . . . . . 9 ({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
76biimpri 218 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ≠ ∅)
8 infssuzcl 11757 . . . . . . . 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 3593 . . . . . 6 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁))
11 nfrab1 3117 . . . . . . . . . . 11 𝑛{𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}
12 nfcv 2762 . . . . . . . . . . 11 𝑛
13 nfcv 2762 . . . . . . . . . . 11 𝑛 <
1411, 12, 13nfinf 8373 . . . . . . . . . 10 𝑛inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )
15 nfcv 2762 . . . . . . . . . 10 𝑛(1..^𝑁)
1614nfcsb1 3541 . . . . . . . . . . 11 𝑛inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
1716nfcri 2756 . . . . . . . . . 10 𝑛 𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
18 csbeq1a 3535 . . . . . . . . . . 11 (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝐴 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
1918eleq2d 2685 . . . . . . . . . 10 (𝑛 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (𝑥𝐴𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
2014, 15, 17, 19elrabf 3354 . . . . . . . . 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 208 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) ∧ 𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
2221simprd 479 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑥inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
231, 2sstri 3604 . . . . . . . . . . 11 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ ℕ
24 nnssre 11009 . . . . . . . . . . 11 ℕ ⊆ ℝ
2523, 24sstri 3604 . . . . . . . . . 10 {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ⊆ ℝ
2625, 9sseldi 3593 . . . . . . . . 9 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
2726ltnrd 10156 . . . . . . . 8 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ¬ inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
28 eliun 4515 . . . . . . . . 9 (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵)
2926ad2antrr 761 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
30 elfzouz2 12468 . . . . . . . . . . . . . . . . 17 (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) ∈ (1..^𝑁) → 𝑁 ∈ (ℤ‘inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
31 fzoss2 12480 . . . . . . . . . . . . . . . . 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 3595 . . . . . . . . . . . . . . 15 ((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) → 𝑘 ∈ (1..^𝑁))
3433adantr 481 . . . . . . . . . . . . . 14 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ (1..^𝑁))
352, 34sseldi 3593 . . . . . . . . . . . . 13 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℕ)
3635nnred 11020 . . . . . . . . . . . 12 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℝ)
37 simpr 477 . . . . . . . . . . . . . 14 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑥𝐵)
38 nfcv 2762 . . . . . . . . . . . . . . 15 𝑛𝑘
39 iundisj3.0 . . . . . . . . . . . . . . . 16 𝑛𝐵
4039nfcri 2756 . . . . . . . . . . . . . . 15 𝑛 𝑥𝐵
41 iundisj3.1 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘𝐴 = 𝐵)
4241eleq2d 2685 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝑥𝐴𝑥𝐵))
4338, 15, 40, 42elrabf 3354 . . . . . . . . . . . . . 14 (𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴} ↔ (𝑘 ∈ (1..^𝑁) ∧ 𝑥𝐵))
4434, 37, 43sylanbrc 697 . . . . . . . . . . . . 13 (((∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ {𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴})
45 infssuzle 11756 . . . . . . . . . . . . 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 12463 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))
4847ad2antlr 762 . . . . . . . . . . . 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 10180 . . . . . . . . . . 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 450 . . . . . . . . . 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 3027 . . . . . . . . 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 232 . . . . . . . 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 189 . . . . . . 7 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ¬ 𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)
5422, 53eldifd 3578 . . . . . 6 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴𝑥 ∈ (inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵))
55 csbeq1 3529 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝑚 / 𝑛𝐴 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
56 oveq2 6643 . . . . . . . . . 10 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < )))
5756iuneq1d 4536 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ) → 𝑘 ∈ (1..^𝑚)𝐵 = 𝑘 ∈ (1..^inf({𝑛 ∈ (1..^𝑁) ∣ 𝑥𝐴}, ℝ, < ))𝐵)
5855, 57difeq12d 3721 . . . . . . . 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 2685 . . . . . . 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 3304 . . . . . 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 692 . . . . 5 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
62 nfv 1841 . . . . . 6 𝑚 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
63 nfcsb1v 3542 . . . . . . . 8 𝑛𝑚 / 𝑛𝐴
64 nfcv 2762 . . . . . . . . 9 𝑛(1..^𝑚)
6564, 39nfiun 4539 . . . . . . . 8 𝑛 𝑘 ∈ (1..^𝑚)𝐵
6663, 65nfdif 3723 . . . . . . 7 𝑛(𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
6766nfcri 2756 . . . . . 6 𝑛 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
68 csbeq1a 3535 . . . . . . . 8 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
69 oveq2 6643 . . . . . . . . 9 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
7069iuneq1d 4536 . . . . . . . 8 (𝑛 = 𝑚 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑚)𝐵)
7168, 70difeq12d 3721 . . . . . . 7 (𝑛 = 𝑚 → (𝐴 𝑘 ∈ (1..^𝑛)𝐵) = (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
7271eleq2d 2685 . . . . . 6 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)))
7362, 67, 72cbvrex 3163 . . . . 5 (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑚 ∈ (1..^𝑁)𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
7461, 73sylibr 224 . . . 4 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 → ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
75 eldifi 3724 . . . . 5 (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥𝐴)
7675reximi 3008 . . . 4 (∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
7774, 76impbii 199 . . 3 (∃𝑛 ∈ (1..^𝑁)𝑥𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
78 eliun 4515 . . 3 (𝑥 𝑛 ∈ (1..^𝑁)𝐴 ↔ ∃𝑛 ∈ (1..^𝑁)𝑥𝐴)
79 eliun 4515 . . 3 (𝑥 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑛 ∈ (1..^𝑁)𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
8077, 78, 793bitr4i 292 . 2 (𝑥 𝑛 ∈ (1..^𝑁)𝐴𝑥 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵))
8180eqriv 2617 1 𝑛 ∈ (1..^𝑁)𝐴 = 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wnfc 2749  wne 2791  wrex 2910  {crab 2913  csb 3526  cdif 3564  wss 3567  c0 3907   ciun 4511   class class class wbr 4644  cfv 5876  (class class class)co 6635  infcinf 8332  cr 9920  1c1 9922   < clt 10059  cle 10060  cn 11005  cuz 11672  ..^cfzo 12449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-fzo 12450
This theorem is referenced by:  iundisjcnt  29531
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