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Theorem ctinfom 11975
 Description: A condition for a set being countably infinite. Restates ennnfone 11972 in terms of ω and function image. Like ennnfone 11972 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 11972 . . . 4 (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖))))
21simplbi 272 . . 3 (𝐴 ≈ ℕ → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
3 nnenom 10237 . . . . . . 7 ℕ ≈ ω
4 entr 6685 . . . . . . 7 ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
53, 4mpan2 422 . . . . . 6 (𝐴 ≈ ℕ → 𝐴 ≈ ω)
65ensymd 6684 . . . . 5 (𝐴 ≈ ℕ → ω ≈ 𝐴)
7 bren 6648 . . . . 5 (ω ≈ 𝐴 ↔ ∃𝑓 𝑓:ω–1-1-onto𝐴)
86, 7sylib 121 . . . 4 (𝐴 ≈ ℕ → ∃𝑓 𝑓:ω–1-1-onto𝐴)
9 f1ofo 5381 . . . . . . . 8 (𝑓:ω–1-1-onto𝐴𝑓:ω–onto𝐴)
109adantl 275 . . . . . . 7 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → 𝑓:ω–onto𝐴)
11 simpr 109 . . . . . . . . 9 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
12 nnord 4532 . . . . . . . . . . . 12 (𝑛 ∈ ω → Ord 𝑛)
1312adantl 275 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14 ordirr 4464 . . . . . . . . . . 11 (Ord 𝑛 → ¬ 𝑛𝑛)
1513, 14syl 14 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ 𝑛𝑛)
16 f1of1 5373 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto𝐴𝑓:ω–1-1𝐴)
1716ad2antlr 481 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–1-1𝐴)
18 omelon 4529 . . . . . . . . . . . . 13 ω ∈ On
1918onelssi 4358 . . . . . . . . . . . 12 (𝑛 ∈ ω → 𝑛 ⊆ ω)
2019adantl 275 . . . . . . . . . . 11 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21 f1elima 5681 . . . . . . . . . . 11 ((𝑓:ω–1-1𝐴𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2217, 11, 20, 21syl3anc 1217 . . . . . . . . . 10 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ((𝑓𝑛) ∈ (𝑓𝑛) ↔ 𝑛𝑛))
2315, 22mtbird 663 . . . . . . . . 9 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓𝑛) ∈ (𝑓𝑛))
24 fveq2 5428 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑓𝑘) = (𝑓𝑛))
2524eleq1d 2209 . . . . . . . . . . 11 (𝑘 = 𝑛 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑛) ∈ (𝑓𝑛)))
2625notbid 657 . . . . . . . . . 10 (𝑘 = 𝑛 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑛) ∈ (𝑓𝑛)))
2726rspcev 2792 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ¬ (𝑓𝑛) ∈ (𝑓𝑛)) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2811, 23, 27syl2anc 409 . . . . . . . 8 (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) ∧ 𝑛 ∈ ω) → ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
2928ralrimiva 2508 . . . . . . 7 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))
3010, 29jca 304 . . . . . 6 ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto𝐴) → (𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)))
3130ex 114 . . . . 5 (𝐴 ≈ ℕ → (𝑓:ω–1-1-onto𝐴 → (𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
3231eximdv 1853 . . . 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 304 . 2 (𝐴 ≈ ℕ → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))))
35 oveq1 5788 . . . . . . . . 9 (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
3635cbvmptv 4031 . . . . . . . 8 (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37 freceq1 6296 . . . . . . . 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 2140 . . . . . . 7 (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))
40 simpl 108 . . . . . . 7 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → 𝑓:ω–onto𝐴)
41 fveq2 5428 . . . . . . . . . . . . 13 (𝑘 = 𝑑 → (𝑓𝑘) = (𝑓𝑑))
4241eleq1d 2209 . . . . . . . . . . . 12 (𝑘 = 𝑑 → ((𝑓𝑘) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑛)))
4342notbid 657 . . . . . . . . . . 11 (𝑘 = 𝑑 → (¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑛)))
4443cbvrexv 2658 . . . . . . . . . 10 (∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
4544ralbii 2444 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛))
46 imaeq2 4884 . . . . . . . . . . . . 13 (𝑛 = 𝑐 → (𝑓𝑛) = (𝑓𝑐))
4746eleq2d 2210 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ((𝑓𝑑) ∈ (𝑓𝑛) ↔ (𝑓𝑑) ∈ (𝑓𝑐)))
4847notbid 657 . . . . . . . . . . 11 (𝑛 = 𝑐 → (¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
4948rexbidv 2439 . . . . . . . . . 10 (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐)))
5049cbvralv 2657 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑛) ↔ ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5145, 50sylbb 122 . . . . . . . 8 (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5251adantl 275 . . . . . . 7 ((𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓𝑑) ∈ (𝑓𝑐))
5338, 39, 40, 52ctinfomlemom 11974 . . . . . 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 2692 . . . . . . . 8 𝑓 ∈ V
55 frecex 6298 . . . . . . . . 9 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5655cnvex 5084 . . . . . . . 8 frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
5754, 56coex 5091 . . . . . . 7 (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58 foeq1 5348 . . . . . . . 8 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔:ℕ0onto𝐴 ↔ (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0onto𝐴))
59 fveq1 5427 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑗) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60 fveq1 5427 . . . . . . . . . . . 12 (𝑔 = (𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔𝑖) = ((𝑓frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
6159, 60neeq12d 2329 . . . . . . . . . . 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 2438 . . . . . . . . . 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 2439 . . . . . . . . 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 2438 . . . . . . . 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 465 . . . . . . 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 2783 . . . . . 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 1578 . . . 4 (∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛)) → ∃𝑔(𝑔:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝑔𝑗) ≠ (𝑔𝑖)))
6968anim2i 340 . . 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 133 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓𝑘) ∈ (𝑓𝑛))) → 𝐴 ≈ ℕ)
7134, 70impbii 125 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 103   ↔ wb 104  DECID wdc 820   = wceq 1332  ∃wex 1469   ∈ wcel 1481   ≠ wne 2309  ∀wral 2417  ∃wrex 2418   ⊆ wss 3075   class class class wbr 3936   ↦ cmpt 3996  Ord word 4291  ωcom 4511  ◡ccnv 4545   “ cima 4549   ∘ ccom 4550  –1-1→wf1 5127  –onto→wfo 5128  –1-1-onto→wf1o 5129  ‘cfv 5130  (class class class)co 5781  freccfrec 6294   ≈ cen 6639  0cc0 7643  1c1 7644   + caddc 7646  ℕcn 8743  ℕ0cn0 9000  ℤcz 9077  ...cfz 9820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-ltadd 7759 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-er 6436  df-pm 6552  df-en 6642  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-fz 9821  df-seqfrec 10249 This theorem is referenced by:  ctinf  11977
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