Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnfnei Structured version   Visualization version   GIF version

Theorem pnfnei 20834
 Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 22417 (which describes neighborhoods of ℝ) and mnfnei 20835, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 20831 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 2610 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2610 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
3 eqid 2610 . . . 4 ran (,) = ran (,)
41, 2, 3leordtval 20827 . . 3 (ordTop‘ ≤ ) = (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)))
54eleq2i 2680 . 2 (𝐴 ∈ (ordTop‘ ≤ ) ↔ 𝐴 ∈ (topGen‘((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,))))
6 tg2 20580 . . 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 3715 . . . . 5 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) ↔ (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑢 ∈ ran (,)))
8 elun 3715 . . . . . . 7 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ↔ (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))
9 vex 3176 . . . . . . . . . 10 𝑢 ∈ V
10 eqid 2610 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
1110elrnmpt 5293 . . . . . . . . . 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 9975 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
1413a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ ∈ ℝ*)
15 simprl 790 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ∈ ℝ*)
16 0xr 9965 . . . . . . . . . . . . . 14 0 ∈ ℝ*
17 ifcl 4080 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
1815, 16, 17sylancl 693 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ*)
19 pnfxr 9971 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
2019a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ ℝ*)
21 xrmax1 11880 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
2216, 15, 21sylancr 694 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
23 ge0gtmnf 11877 . . . . . . . . . . . . . 14 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
2418, 22, 23syl2anc 691 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → -∞ < if(0 ≤ 𝑦, 𝑦, 0))
25 simpll 786 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ 𝑢)
26 simprr 792 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢 = (𝑦(,]+∞))
2725, 26eleqtrd 2690 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → +∞ ∈ (𝑦(,]+∞))
28 elioc1 12088 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
2915, 19, 28sylancl 693 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞)))
3027, 29mpbid 221 . . . . . . . . . . . . . . 15 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (+∞ ∈ ℝ*𝑦 < +∞ ∧ +∞ ≤ +∞))
3130simp2d 1067 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 < +∞)
32 0ltpnf 11832 . . . . . . . . . . . . . 14 0 < +∞
33 breq1 4586 . . . . . . . . . . . . . . 15 (𝑦 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑦 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
34 breq1 4586 . . . . . . . . . . . . . . 15 (0 = if(0 ≤ 𝑦, 𝑦, 0) → (0 < +∞ ↔ if(0 ≤ 𝑦, 𝑦, 0) < +∞))
3533, 34ifboth 4074 . . . . . . . . . . . . . 14 ((𝑦 < +∞ ∧ 0 < +∞) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
3631, 32, 35sylancl 693 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) < +∞)
37 xrre2 11875 . . . . . . . . . . . . 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 1326 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
39 xrmax2 11881 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ*𝑦 ∈ ℝ*) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
4016, 15, 39sylancr 694 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0))
41 df-ioc 12051 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
42 xrlelttr 11863 . . . . . . . . . . . . . . 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 12064 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ*𝑦 ≤ if(0 ≤ 𝑦, 𝑦, 0)) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
4415, 40, 43syl2anc 691 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ (𝑦(,]+∞))
45 simplr 788 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → 𝑢𝐴)
4626, 45eqsstr3d 3603 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (𝑦(,]+∞) ⊆ 𝐴)
4744, 46sstrd 3578 . . . . . . . . . . . 12 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴)
48 oveq1 6556 . . . . . . . . . . . . . 14 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → (𝑥(,]+∞) = (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞))
4948sseq1d 3595 . . . . . . . . . . . . 13 (𝑥 = if(0 ≤ 𝑦, 𝑦, 0) → ((𝑥(,]+∞) ⊆ 𝐴 ↔ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴))
5049rspcev 3282 . . . . . . . . . . . 12 ((if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≤ 𝑦, 𝑦, 0)(,]+∞) ⊆ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5138, 47, 50syl2anc 691 . . . . . . . . . . 11 (((+∞ ∈ 𝑢𝑢𝐴) ∧ (𝑦 ∈ ℝ*𝑢 = (𝑦(,]+∞))) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
5251rexlimdvaa 3014 . . . . . . . . . 10 ((+∞ ∈ 𝑢𝑢𝐴) → (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5352com12 32 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (𝑦(,]+∞) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
5412, 53sylbi 206 . . . . . . . 8 (𝑢 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
55 eqid 2610 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
5655elrnmpt 5293 . . . . . . . . . 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 11838 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
59 elico1 12089 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*) → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
6013, 59mpan 702 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦)))
61 simp3 1056 . . . . . . . . . . . . . . 15 ((+∞ ∈ ℝ* ∧ -∞ ≤ +∞ ∧ +∞ < 𝑦) → +∞ < 𝑦)
6260, 61syl6bi 242 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → (+∞ ∈ (-∞[,)𝑦) → +∞ < 𝑦))
6358, 62mtod 188 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ¬ +∞ ∈ (-∞[,)𝑦))
64 eleq2 2677 . . . . . . . . . . . . . 14 (𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (-∞[,)𝑦)))
6564notbid 307 . . . . . . . . . . . . 13 (𝑢 = (-∞[,)𝑦) → (¬ +∞ ∈ 𝑢 ↔ ¬ +∞ ∈ (-∞[,)𝑦)))
6663, 65syl5ibrcom 236 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢))
6766rexlimiv 3009 . . . . . . . . . . 11 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ¬ +∞ ∈ 𝑢)
6867pm2.21d 117 . . . . . . . . . 10 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
6968adantrd 483 . . . . . . . . 9 (∃𝑦 ∈ ℝ* 𝑢 = (-∞[,)𝑦) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
7057, 69sylbi 206 . . . . . . . 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 206 . . . . . 6 (𝑢 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
73 pnfnre 9960 . . . . . . . . . 10 +∞ ∉ ℝ
7473neli 2885 . . . . . . . . 9 ¬ +∞ ∈ ℝ
75 elssuni 4403 . . . . . . . . . . 11 (𝑢 ∈ ran (,) → 𝑢 ran (,))
76 unirnioo 12144 . . . . . . . . . . 11 ℝ = ran (,)
7775, 76syl6sseqr 3615 . . . . . . . . . 10 (𝑢 ∈ ran (,) → 𝑢 ⊆ ℝ)
7877sseld 3567 . . . . . . . . 9 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → +∞ ∈ ℝ))
7974, 78mtoi 189 . . . . . . . 8 (𝑢 ∈ ran (,) → ¬ +∞ ∈ 𝑢)
8079pm2.21d 117 . . . . . . 7 (𝑢 ∈ ran (,) → (+∞ ∈ 𝑢 → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8180adantrd 483 . . . . . 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 206 . . . 4 (𝑢 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∪ ran (,)) → ((+∞ ∈ 𝑢𝑢𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴))
8483rexlimiv 3009 . . 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 488 1 ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ifcif 4036  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  (,)cioo 12046  (,]cioc 12047  [,)cico 12048  topGenctg 15921  ordTopcordt 15982 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-topgen 15927  df-ordt 15984  df-ps 17023  df-tsr 17024  df-top 20521  df-bases 20522 This theorem is referenced by:  xrge0tsms  22445  xrlimcnp  24495  xrge0tsmsd  29116  pnfneige0  29325
 Copyright terms: Public domain W3C validator