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Theorem ctinfom 12428
Description: A condition for a set being countably infinite. Restates ennnfone 12425 in terms of ω and function image. Like ennnfone 12425 the condition can be summarized as 𝐴 being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)
Assertion
Ref Expression
ctinfom (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
Distinct variable groups:   𝐴,𝑓,𝑛   𝑥,𝐴,𝑦   𝑓,𝑘,𝑛
Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem ctinfom
Dummy variables 𝑎 𝑑 𝑖 𝑚 𝑔 𝑏 𝑐 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ennnfone 12425 . . . 4 (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖))))
21simplbi 274 . . 3 (𝐴 ≈ ℕ → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
3 nnenom 10433 . . . . . . 7 ℕ ≈ ω
4 entr 6783 . . . . . . 7 ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
53, 4mpan2 425 . . . . . 6 (𝐴 ≈ ℕ → 𝐴 ≈ ω)
65ensymd 6782 . . . . 5 (𝐴 ≈ ℕ → ω ≈ 𝐴)
7 bren 6746 . . . . 5 (ω ≈ 𝐴 ↔ ∃𝑓 𝑓:ω–1-1-onto𝐴)
86, 7sylib 122 . . . 4 (𝐴 ≈ ℕ → ∃𝑓 𝑓:ω–1-1-onto𝐴)
9 f1ofo 5468 . . . . . . . 8 (𝑓:ω–1-1-onto𝐴𝑓:ω–onto𝐴)
109adantl 277 . . . . . . 7 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → 𝑓:ω–onto𝐴)
11 simpr 110 . . . . . . . . 9 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
12 nnord 4611 . . . . . . . . . . . 12 (𝑛 ∈ ω → Ord 𝑛)
1312adantl 277 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14 ordirr 4541 . . . . . . . . . . 11 (Ord 𝑛 → ¬ 𝑛𝑛)
1513, 14syl 14 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ 𝑛𝑛)
16 f1of1 5460 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto𝐴𝑓:ω–1-1𝐴)
1716ad2antlr 489 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–1-1𝐴)
18 omelon 4608 . . . . . . . . . . . . 13 ω ∈ On
1918onelssi 4429 . . . . . . . . . . . 12 (𝑛 ∈ ω → 𝑛 ⊆ ω)
2019adantl 277 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21 f1elima 5773 . . . . . . . . . . 11 ((𝑓:ω–1-1𝐴𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2217, 11, 20, 21syl3anc 1238 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2315, 22mtbird 673 . . . . . . . . 9 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓𝑛) ∈ (𝑓𝑛))
24 fveq2 5515 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑓𝑘) = (𝑓𝑛))
2524eleq1d 2246 . . . . . . . . . . 11 (𝑘 = 𝑛 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑛) ∈ (𝑓𝑛)))
2625notbid 667 . . . . . . . . . 10 (𝑘 = 𝑛 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑛) ∈ (𝑓𝑛)))
2726rspcev 2841 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ¬ (𝑓𝑛) ∈ (𝑓𝑛)) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2811, 23, 27syl2anc 411 . . . . . . . 8 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2928ralrimiva 2550 . . . . . . 7 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
3010, 29jca 306 . . . . . 6 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → (𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)))
3130ex 115 . . . . 5 (𝐴 ≈ ℕ → (𝑓:ω–1-1-onto𝐴 → (𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
3231eximdv 1880 . . . 4 (𝐴 ≈ ℕ → (∃𝑓 𝑓:ω–1-1-onto𝐴 → ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
338, 32mpd 13 . . 3 (𝐴 ≈ ℕ → ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)))
342, 33jca 306 . 2 (𝐴 ≈ ℕ → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
35 oveq1 5881 . . . . . . . . 9 (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
3635cbvmptv 4099 . . . . . . . 8 (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37 freceq1 6392 . . . . . . . 8 ((𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1)) → frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0))
3836, 37ax-mp 5 . . . . . . 7 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0)
39 eqid 2177 . . . . . . 7 (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))
40 simpl 109 . . . . . . 7 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → 𝑓:ω–onto𝐴)
41 fveq2 5515 . . . . . . . . . . . . 13 (𝑘 = 𝑑 → (𝑓𝑘) = (𝑓𝑑))
4241eleq1d 2246 . . . . . . . . . . . 12 (𝑘 = 𝑑 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑛)))
4342notbid 667 . . . . . . . . . . 11 (𝑘 = 𝑑 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑛)))
4443cbvrexv 2704 . . . . . . . . . 10 (∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
4544ralbii 2483 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
46 imaeq2 4966 . . . . . . . . . . . . 13 (𝑛 = 𝑐 → (𝑓𝑛) = (𝑓𝑐))
4746eleq2d 2247 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ((𝑓𝑑) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑐)))
4847notbid 667 . . . . . . . . . . 11 (𝑛 = 𝑐 → (¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
4948rexbidv 2478 . . . . . . . . . 10 (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
5049cbvralv 2703 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5145, 50sylbb 123 . . . . . . . 8 (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5251adantl 277 . . . . . . 7 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5338, 39, 40, 52ctinfomlemom 12427 . . . . . 6 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
54 vex 2740 . . . . . . . 8 𝑓 ∈ V
55 frecex 6394 . . . . . . . . 9 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5655cnvex 5167 . . . . . . . 8 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5754, 56coex 5174 . . . . . . 7 (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58 foeq1 5434 . . . . . . . 8 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔:ℕ0onto𝐴 ↔ (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴))
59 fveq1 5514 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑗) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60 fveq1 5514 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑖) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
6159, 60neeq12d 2367 . . . . . . . . . . 11 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → ((𝑔𝑗) ≠ (𝑔𝑖) ↔ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6261ralbidv 2477 . . . . . . . . . 10 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∀𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖) ↔ ∀𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6362rexbidv 2478 . . . . . . . . 9 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖) ↔ ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6463ralbidv 2477 . . . . . . . 8 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖) ↔ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
6558, 64anbi12d 473 . . . . . . 7 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → ((𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)) ↔ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))))
6657, 65spcev 2832 . . . . . 6 (((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6753, 66syl 14 . . . . 5 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6867exlimiv 1598 . . . 4 (∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6968anim2i 342 . . 3 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))) → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖))))
7069, 1sylibr 134 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))) → 𝐴 ≈ ℕ)
7134, 70impbii 126 1 (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  DECID wdc 834   = wceq 1353  wex 1492  wcel 2148  wne 2347  wral 2455  wrex 2456  wss 3129   class class class wbr 4003  cmpt 4064  Ord word 4362  ωcom 4589  ccnv 4625  cima 4629  ccom 4630  1-1wf1 5213  ontowfo 5214  1-1-ontowf1o 5215  cfv 5216  (class class class)co 5874  freccfrec 6390  cen 6737  0cc0 7810  1c1 7811   + caddc 7813  cn 8918  0cn0 9175  cz 9252  ...cfz 10007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-er 6534  df-pm 6650  df-en 6740  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-fz 10008  df-seqfrec 10445
This theorem is referenced by:  ctinf  12430
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