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Theorem pnfnei 21829
Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 23415 (which describes neighborhoods of ) and mnfnei 21830, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 21826 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 2801 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2801 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
3 eqid 2801 . . . 4 ran (,) = ran (,)
41, 2, 3leordtval 21822 . . 3 (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)))
54eleq2i 2884 . 2 (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6 tg2 21574 . . 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 4079 . . . . 5 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8 elun 4079 . . . . . . 7 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))
9 eqid 2801 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
109elrnmpt 5796 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞)))
1110elv 3449 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞))
12 mnfxr 10691 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
1312a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ*)
14 simprl 770 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ*)
15 0xr 10681 . . . . . . . . . . . . . 14 0 ∈ ℝ*
16 ifcl 4472 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
1714, 15, 16sylancl 589 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
18 pnfxr 10688 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
1918a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ*)
20 xrmax1 12560 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
2115, 14, 20sylancr 590 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
22 ge0gtmnf 12557 . . . . . . . . . . . . . 14 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
2317, 21, 22syl2anc 587 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
24 simpll 766 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
25 simprr 772 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
2624, 25eleqtrd 2895 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
27 elioc1 12772 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2814, 18, 27sylancl 589 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2926, 28mpbid 235 . . . . . . . . . . . . . . 15 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞))
3029simp2d 1140 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
31 0ltpnf 12509 . . . . . . . . . . . . . 14 0 < +∞
32 breq1 5036 . . . . . . . . . . . . . . 15 (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
33 breq1 5036 . . . . . . . . . . . . . . 15 (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
3432, 33ifboth 4466 . . . . . . . . . . . . . 14 ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
3530, 31, 34sylancl 589 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
36 xrre2 12555 . . . . . . . . . . . . 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 1375 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
38 xrmax2 12561 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
3915, 14, 38sylancr 590 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
40 df-ioc 12735 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
41 xrlelttr 12541 . . . . . . . . . . . . . . 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 12748 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ*𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
4314, 39, 42syl2anc 587 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
44 simplr 768 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢𝐴)
4525, 44eqsstrrd 3957 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
4643, 45sstrd 3928 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
47 oveq1 7146 . . . . . . . . . . . . . 14 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
4847sseq1d 3949 . . . . . . . . . . . . 13 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
4948rspcev 3574 . . . . . . . . . . . 12 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5037, 46, 49syl2anc 587 . . . . . . . . . . 11 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5150rexlimdvaa 3247 . . . . . . . . . 10 ((+∞ ∈ 𝑢𝑢𝐴) → (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5251com12 32 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5311, 52sylbi 220 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
54 eqid 2801 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
5554elrnmpt 5796 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦)))
5655elv 3449 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦))
57 pnfnlt 12515 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
58 elico1 12773 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
5912, 58mpan 689 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
60 simp3 1135 . . . . . . . . . . . . . . 15 ((+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
6159, 60syl6bi 256 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
6257, 61mtod 201 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ¬ +∞ ∈ (-∞[,)𝑦))
63 eleq2 2881 . . . . . . . . . . . . . 14 (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
6463notbid 321 . . . . . . . . . . . . 13 (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
6562, 64syl5ibrcom 250 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
6665rexlimiv 3242 . . . . . . . . . . 11 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
6766pm2.21d 121 . . . . . . . . . 10 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6867adantrd 495 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6956, 68sylbi 220 . . . . . . . 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 220 . . . . . 6 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
72 pnfnre 10675 . . . . . . . . . 10 +∞ ∉ ℝ
7372neli 3096 . . . . . . . . 9 ¬ +∞ ∈ ℝ
74 elssuni 4833 . . . . . . . . . . 11 (𝑢 ∈ ran (,) → 𝑢 ran (,))
75 unirnioo 12831 . . . . . . . . . . 11 ℝ = ran (,)
7674, 75sseqtrrdi 3969 . . . . . . . . . 10 (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
7776sseld 3917 . . . . . . . . 9 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
7873, 77mtoi 202 . . . . . . . 8 (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
7978pm2.21d 121 . . . . . . 7 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8079adantrd 495 . . . . . 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 220 . . . 4 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8382rexlimiv 3242 . . 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 584 1 ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wrex 3110  Vcvv 3444  cun 3882  wss 3884  ifcif 4428   cuni 4803   class class class wbr 5033  cmpt 5113  ran crn 5524  cfv 6328  (class class class)co 7139  cr 10529  0cc0 10530  +∞cpnf 10665  -∞cmnf 10666  *cxr 10667   < clt 10668  cle 10669  (,)cioo 12730  (,]cioc 12731  [,)cico 12732  topGenctg 16707  ordTopcordt 16768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-topgen 16713  df-ordt 16770  df-ps 17806  df-tsr 17807  df-top 21503  df-bases 21555
This theorem is referenced by:  xrge0tsms  23443  xrlimcnp  25558  xrge0tsmsd  30746  pnfneige0  31308  xlimpnfvlem2  42476
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