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Theorem pnfnei 23243
Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24841 (which describes neighborhoods of ) and mnfnei 23244, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 23240 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 2734 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2734 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
3 eqid 2734 . . . 4 ran (,) = ran (,)
41, 2, 3leordtval 23236 . . 3 (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)))
54eleq2i 2830 . 2 (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6 tg2 22987 . . 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 4162 . . . . 5 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8 elun 4162 . . . . . . 7 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))
9 eqid 2734 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
109elrnmpt 5971 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞)))
1110elv 3482 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞))
12 mnfxr 11315 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
1312a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ*)
14 simprl 771 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ*)
15 0xr 11305 . . . . . . . . . . . . . 14 0 ∈ ℝ*
16 ifcl 4575 . . . . . . . . . . . . . 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 11312 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
1918a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ*)
20 xrmax1 13213 . . . . . . . . . . . . . . 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 13210 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25 simprr 773 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
2624, 25eleqtrd 2840 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27 elioc1 13425 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2814, 18, 27sylancl 586 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2926, 28mpbid 232 . . . . . . . . . . . . . . 15 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞))
3029simp2d 1142 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31 0ltpnf 13161 . . . . . . . . . . . . . 14 0 < +∞
32 breq1 5150 . . . . . . . . . . . . . . 15 (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33 breq1 5150 . . . . . . . . . . . . . . 15 (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
3432, 33ifboth 4569 . . . . . . . . . . . . . 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 13208 . . . . . . . . . . . . 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 13214 . . . . . . . . . . . . . . 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 13388 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
41 xrlelttr 13194 . . . . . . . . . . . . . . 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 13401 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢𝐴)
4525, 44eqsstrrd 4034 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
4643, 45sstrd 4005 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47 oveq1 7437 . . . . . . . . . . . . . 14 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
4847sseq1d 4026 . . . . . . . . . . . . 13 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
4948rspcev 3621 . . . . . . . . . . . 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 3153 . . . . . . . . . 10 ((+∞ ∈ 𝑢𝑢𝐴) → (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5251com12 32 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5311, 52sylbi 217 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
54 eqid 2734 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
5554elrnmpt 5971 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦)))
5655elv 3482 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦))
57 pnfnlt 13167 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
58 elico1 13426 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
5912, 58mpan 690 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60 simp3 1137 . . . . . . . . . . . . . . 15 ((+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
6159, 60biimtrdi 253 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
6257, 61mtod 198 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ¬ +∞ ∈ (-∞[,)𝑦))
63 eleq2 2827 . . . . . . . . . . . . . 14 (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
6463notbid 318 . . . . . . . . . . . . 13 (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
6562, 64syl5ibrcom 247 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
6665rexlimiv 3145 . . . . . . . . . . 11 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
6766pm2.21d 121 . . . . . . . . . 10 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6867adantrd 491 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6956, 68sylbi 217 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
7053, 69jaoi 857 . . . . . . 7 ((𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
718, 70sylbi 217 . . . . . 6 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72 pnfnre 11299 . . . . . . . . . 10 +∞ ∉ ℝ
7372neli 3045 . . . . . . . . 9 ¬ +∞ ∈ ℝ
74 elssuni 4941 . . . . . . . . . . 11 (𝑢 ∈ ran (,) → 𝑢 ran (,))
75 unirnioo 13485 . . . . . . . . . . 11 ℝ = ran (,)
7674, 75sseqtrrdi 4046 . . . . . . . . . 10 (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
7776sseld 3993 . . . . . . . . 9 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
7873, 77mtoi 199 . . . . . . . 8 (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
7978pm2.21d 121 . . . . . . 7 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8079adantrd 491 . . . . . 6 (𝑢 ∈ ran (,) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8171, 80jaoi 857 . . . . 5 ((𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
827, 81sylbi 217 . . . 4 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8382rexlimiv 3145 . . 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  cun 3960  wss 3962  ifcif 4530   cuni 4911   class class class wbr 5147  cmpt 5230  ran crn 5689  cfv 6562  (class class class)co 7430  cr 11151  0cc0 11152  +∞cpnf 11289  -∞cmnf 11290  *cxr 11291   < clt 11292  cle 11293  (,)cioo 13383  (,]cioc 13384  [,)cico 13385  topGenctg 17483  ordTopcordt 17545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-topgen 17489  df-ordt 17547  df-ps 18623  df-tsr 18624  df-top 22915  df-bases 22968
This theorem is referenced by:  xrge0tsms  24869  xrlimcnp  27025  xrge0tsmsd  33047  pnfneige0  33911  xlimpnfvlem2  45792
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