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Theorem pnfnei 22477
Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24075 (which describes neighborhoods of ) and mnfnei 22478, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 22474 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
pnfnei ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem pnfnei
Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2736 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
3 eqid 2736 . . . 4 ran (,) = ran (,)
41, 2, 3leordtval 22470 . . 3 (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)))
54eleq2i 2828 . 2 (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6 tg2 22221 . . 3 ((𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))) ∧ +∞ ∈ 𝐴) → ∃𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))(+∞ ∈ 𝑢𝑢𝐴))
7 elun 4095 . . . . 5 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8 elun 4095 . . . . . . 7 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))
9 eqid 2736 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
109elrnmpt 5897 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞)))
1110elv 3447 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞))
12 mnfxr 11133 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
1312a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ*)
14 simprl 768 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ*)
15 0xr 11123 . . . . . . . . . . . . . 14 0 ∈ ℝ*
16 ifcl 4518 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
1714, 15, 16sylancl 586 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
18 pnfxr 11130 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
1918a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ*)
20 xrmax1 13010 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
2115, 14, 20sylancr 587 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
22 ge0gtmnf 13007 . . . . . . . . . . . . . 14 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
2317, 21, 22syl2anc 584 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
24 simpll 764 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25 simprr 770 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
2624, 25eleqtrd 2839 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27 elioc1 13222 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2814, 18, 27sylancl 586 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2926, 28mpbid 231 . . . . . . . . . . . . . . 15 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞))
3029simp2d 1142 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31 0ltpnf 12959 . . . . . . . . . . . . . 14 0 < +∞
32 breq1 5095 . . . . . . . . . . . . . . 15 (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33 breq1 5095 . . . . . . . . . . . . . . 15 (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
3432, 33ifboth 4512 . . . . . . . . . . . . . 14 ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
3530, 31, 34sylancl 586 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
36 xrre2 13005 . . . . . . . . . . . . 13 (((-∞ ∈ ℝ* ∧ if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < if(0 ≤ 𝑦, 𝑦, 0) ∧ if(0 ≤ 𝑦, 𝑦, 0) < +∞)) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
3713, 17, 19, 23, 35, 36syl32anc 1377 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
38 xrmax2 13011 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
3915, 14, 38sylancr 587 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
40 df-ioc 13185 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
41 xrlelttr 12991 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*𝑥 ∈ ℝ*) → ((𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0) ∧ if(0 ≤ 𝑦, 𝑦, 0) < 𝑥) → 𝑦 < 𝑥))
4240, 40, 41ixxss1 13198 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ*𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
4314, 39, 42syl2anc 584 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
44 simplr 766 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢𝐴)
4525, 44eqsstrrd 3971 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
4643, 45sstrd 3942 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47 oveq1 7344 . . . . . . . . . . . . . 14 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
4847sseq1d 3963 . . . . . . . . . . . . 13 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
4948rspcev 3570 . . . . . . . . . . . 12 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5037, 46, 49syl2anc 584 . . . . . . . . . . 11 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5150rexlimdvaa 3149 . . . . . . . . . 10 ((+∞ ∈ 𝑢𝑢𝐴) → (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5251com12 32 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5311, 52sylbi 216 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
54 eqid 2736 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
5554elrnmpt 5897 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦)))
5655elv 3447 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦))
57 pnfnlt 12965 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
58 elico1 13223 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
5912, 58mpan 687 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60 simp3 1137 . . . . . . . . . . . . . . 15 ((+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
6159, 60syl6bi 252 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
6257, 61mtod 197 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ¬ +∞ ∈ (-∞[,)𝑦))
63 eleq2 2825 . . . . . . . . . . . . . 14 (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
6463notbid 317 . . . . . . . . . . . . 13 (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
6562, 64syl5ibrcom 246 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
6665rexlimiv 3141 . . . . . . . . . . 11 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
6766pm2.21d 121 . . . . . . . . . 10 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6867adantrd 492 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6956, 68sylbi 216 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
7053, 69jaoi 854 . . . . . . 7 ((𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
718, 70sylbi 216 . . . . . 6 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72 pnfnre 11117 . . . . . . . . . 10 +∞ ∉ ℝ
7372neli 3048 . . . . . . . . 9 ¬ +∞ ∈ ℝ
74 elssuni 4885 . . . . . . . . . . 11 (𝑢 ∈ ran (,) → 𝑢 ran (,))
75 unirnioo 13282 . . . . . . . . . . 11 ℝ = ran (,)
7674, 75sseqtrrdi 3983 . . . . . . . . . 10 (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
7776sseld 3931 . . . . . . . . 9 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
7873, 77mtoi 198 . . . . . . . 8 (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
7978pm2.21d 121 . . . . . . 7 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8079adantrd 492 . . . . . 6 (𝑢 ∈ ran (,) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8171, 80jaoi 854 . . . . 5 ((𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
827, 81sylbi 216 . . . 4 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8382rexlimiv 3141 . . 3 (∃𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))(+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
846, 83syl 17 . 2 ((𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
855, 84sylanb 581 1 ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1540  wcel 2105  wrex 3070  Vcvv 3441  cun 3896  wss 3898  ifcif 4473   cuni 4852   class class class wbr 5092  cmpt 5175  ran crn 5621  cfv 6479  (class class class)co 7337  cr 10971  0cc0 10972  +∞cpnf 11107  -∞cmnf 11108  *cxr 11109   < clt 11110  cle 11111  (,)cioo 13180  (,]cioc 13181  [,)cico 13182  topGenctg 17245  ordTopcordt 17307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-1o 8367  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-fi 9268  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-ioo 13184  df-ioc 13185  df-ico 13186  df-icc 13187  df-topgen 17251  df-ordt 17309  df-ps 18381  df-tsr 18382  df-top 22149  df-bases 22202
This theorem is referenced by:  xrge0tsms  24103  xrlimcnp  26224  xrge0tsmsd  31604  pnfneige0  32199  xlimpnfvlem2  43722
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