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Theorem pnfnei 21246
Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 22830 (which describes neighborhoods of ) and mnfnei 21247, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 21243 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 2760 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2760 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
3 eqid 2760 . . . 4 ran (,) = ran (,)
41, 2, 3leordtval 21239 . . 3 (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)))
54eleq2i 2831 . 2 (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6 tg2 20991 . . 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 3896 . . . . 5 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8 elun 3896 . . . . . . 7 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))
9 vex 3343 . . . . . . . . . 10 𝑢 ∈ V
10 eqid 2760 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
1110elrnmpt 5527 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞)))
129, 11ax-mp 5 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞))
13 mnfxr 10308 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
1413a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ*)
15 simprl 811 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ*)
16 0xr 10298 . . . . . . . . . . . . . 14 0 ∈ ℝ*
17 ifcl 4274 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
1815, 16, 17sylancl 697 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
19 pnfxr 10304 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
2019a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ*)
21 xrmax1 12219 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
2216, 15, 21sylancr 698 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
23 ge0gtmnf 12216 . . . . . . . . . . . . . 14 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
2418, 22, 23syl2anc 696 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
25 simpll 807 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
26 simprr 813 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
2725, 26eleqtrd 2841 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
28 elioc1 12430 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2915, 19, 28sylancl 697 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
3027, 29mpbid 222 . . . . . . . . . . . . . . 15 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞))
3130simp2d 1138 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
32 0ltpnf 12169 . . . . . . . . . . . . . 14 0 < +∞
33 breq1 4807 . . . . . . . . . . . . . . 15 (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
34 breq1 4807 . . . . . . . . . . . . . . 15 (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
3533, 34ifboth 4268 . . . . . . . . . . . . . 14 ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
3631, 32, 35sylancl 697 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
37 xrre2 12214 . . . . . . . . . . . . 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)
3814, 18, 20, 24, 36, 37syl32anc 1485 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
39 xrmax2 12220 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
4016, 15, 39sylancr 698 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
41 df-ioc 12393 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
42 xrlelttr 12200 . . . . . . . . . . . . . . 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))
4341, 41, 42ixxss1 12406 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ*𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
4415, 40, 43syl2anc 696 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
45 simplr 809 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢𝐴)
4626, 45eqsstr3d 3781 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
4744, 46sstrd 3754 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
48 oveq1 6821 . . . . . . . . . . . . . 14 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
4948sseq1d 3773 . . . . . . . . . . . . 13 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
5049rspcev 3449 . . . . . . . . . . . 12 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5138, 47, 50syl2anc 696 . . . . . . . . . . 11 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5251rexlimdvaa 3170 . . . . . . . . . 10 ((+∞ ∈ 𝑢𝑢𝐴) → (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5352com12 32 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5412, 53sylbi 207 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
55 eqid 2760 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
5655elrnmpt 5527 . . . . . . . . . 10 (𝑢 ∈ V → (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦)))
579, 56ax-mp 5 . . . . . . . . 9 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ ∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦))
58 pnfnlt 12175 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
59 elico1 12431 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
6013, 59mpan 708 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
61 simp3 1133 . . . . . . . . . . . . . . 15 ((+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
6260, 61syl6bi 243 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
6358, 62mtod 189 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ¬ +∞ ∈ (-∞[,)𝑦))
64 eleq2 2828 . . . . . . . . . . . . . 14 (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
6564notbid 307 . . . . . . . . . . . . 13 (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
6663, 65syl5ibrcom 237 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
6766rexlimiv 3165 . . . . . . . . . . 11 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
6867pm2.21d 118 . . . . . . . . . 10 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6968adantrd 485 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
7057, 69sylbi 207 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
7154, 70jaoi 393 . . . . . . 7 ((𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
728, 71sylbi 207 . . . . . 6 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
73 pnfnre 10293 . . . . . . . . . 10 +∞ ∉ ℝ
7473neli 3037 . . . . . . . . 9 ¬ +∞ ∈ ℝ
75 elssuni 4619 . . . . . . . . . . 11 (𝑢 ∈ ran (,) → 𝑢 ran (,))
76 unirnioo 12486 . . . . . . . . . . 11 ℝ = ran (,)
7775, 76syl6sseqr 3793 . . . . . . . . . 10 (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
7877sseld 3743 . . . . . . . . 9 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
7974, 78mtoi 190 . . . . . . . 8 (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
8079pm2.21d 118 . . . . . . 7 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8180adantrd 485 . . . . . 6 (𝑢 ∈ ran (,) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8272, 81jaoi 393 . . . . 5 ((𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
837, 82sylbi 207 . . . 4 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8483rexlimiv 3165 . . 3 (∃𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))(+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
856, 84syl 17 . 2 ((𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
865, 85sylanb 490 1 ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  cun 3713  wss 3715  ifcif 4230   cuni 4588   class class class wbr 4804  cmpt 4881  ran crn 5267  cfv 6049  (class class class)co 6814  cr 10147  0cc0 10148  +∞cpnf 10283  -∞cmnf 10284  *cxr 10285   < clt 10286  cle 10287  (,)cioo 12388  (,]cioc 12389  [,)cico 12390  topGenctg 16320  ordTopcordt 16381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fi 8484  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-ioo 12392  df-ioc 12393  df-ico 12394  df-icc 12395  df-topgen 16326  df-ordt 16383  df-ps 17421  df-tsr 17422  df-top 20921  df-bases 20972
This theorem is referenced by:  xrge0tsms  22858  xrlimcnp  24915  xrge0tsmsd  30115  pnfneige0  30327  xlimpnfvlem2  40584
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