MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnfnei Structured version   Visualization version   GIF version
Theorem pnfnei 22724
Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24322 (which describes neighborhoods of ℝ) and mnfnei 22725, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 22721 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
pnfnei ((𝐴 ∈ (ordTopβ€˜ ≀ ) ∧ +∞ ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
Distinct variable group:   π‘₯,𝐴
Proof of Theorem pnfnei
Dummy variables π‘Ž 𝑏 𝑐 𝑒 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2733 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
3 eqid 2733 . . . 4 ran (,) = ran (,)
41, 2, 3leordtval 22717 . . 3 (ordTopβ€˜ ≀ ) = (topGenβ€˜((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,)))
54eleq2i 2826 . 2 (𝐴 ∈ (ordTopβ€˜ ≀ ) ↔ 𝐴 ∈ (topGenβ€˜((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,))))
6 tg2 22468 . . 3 ((𝐴 ∈ (topGenβ€˜((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,))) ∧ +∞ ∈ 𝐴) β†’ βˆƒπ‘’ ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,))(+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴))
7 elun 4149 . . . . 5 (𝑒 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,)) ↔ (𝑒 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑒 ∈ ran (,)))
8 elun 4149 . . . . . . 7 (𝑒 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ↔ (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))
9 eqid 2733 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
109elrnmpt 5956 . . . . . . . . . 10 (𝑒 ∈ V β†’ (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (𝑦(,]+∞)))
1110elv 3481 . . . . . . . . 9 (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (𝑦(,]+∞))
12 mnfxr 11271 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
1312a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ -∞ ∈ ℝ*)
14 simprl 770 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑦 ∈ ℝ*)
15 0xr 11261 . . . . . . . . . . . . . 14 0 ∈ ℝ*
16 ifcl 4574 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ*)
1714, 15, 16sylancl 587 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ*)
18 pnfxr 11268 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
1918a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ +∞ ∈ ℝ*)
20 xrmax1 13154 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) β†’ 0 ≀ if(0 ≀ 𝑦, 𝑦, 0))
2115, 14, 20sylancr 588 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 0 ≀ if(0 ≀ 𝑦, 𝑦, 0))
22 ge0gtmnf 13151 . . . . . . . . . . . . . 14 ((if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ* ∧ 0 ≀ if(0 ≀ 𝑦, 𝑦, 0)) β†’ -∞ < if(0 ≀ 𝑦, 𝑦, 0))
2317, 21, 22syl2anc 585 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ -∞ < if(0 ≀ 𝑦, 𝑦, 0))
24 simpll 766 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ +∞ ∈ 𝑒)
25 simprr 772 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑒 = (𝑦(,]+∞))
2624, 25eleqtrd 2836 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ +∞ ∈ (𝑦(,]+∞))
27 elioc1 13366 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) β†’ (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ* ∧ 𝑦 < +∞ ∧ +∞ ≀ +∞)))
2814, 18, 27sylancl 587 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ* ∧ 𝑦 < +∞ ∧ +∞ ≀ +∞)))
2926, 28mpbid 231 . . . . . . . . . . . . . . 15 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (+∞ ∈ ℝ* ∧ 𝑦 < +∞ ∧ +∞ ≀ +∞))
3029simp2d 1144 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑦 < +∞)
31 0ltpnf 13102 . . . . . . . . . . . . . 14 0 < +∞
32 breq1 5152 . . . . . . . . . . . . . . 15 (𝑦 = if(0 ≀ 𝑦, 𝑦, 0) β†’ (𝑦 < +∞ ↔ if(0 ≀ 𝑦, 𝑦, 0) < +∞))
33 breq1 5152 . . . . . . . . . . . . . . 15 (0 = if(0 ≀ 𝑦, 𝑦, 0) β†’ (0 < +∞ ↔ if(0 ≀ 𝑦, 𝑦, 0) < +∞))
3432, 33ifboth 4568 . . . . . . . . . . . . . 14 ((𝑦 < +∞ ∧ 0 < +∞) β†’ if(0 ≀ 𝑦, 𝑦, 0) < +∞)
3530, 31, 34sylancl 587 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ if(0 ≀ 𝑦, 𝑦, 0) < +∞)
36 xrre2 13149 . . . . . . . . . . . . 13 (((-∞ ∈ ℝ* ∧ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < if(0 ≀ 𝑦, 𝑦, 0) ∧ if(0 ≀ 𝑦, 𝑦, 0) < +∞)) β†’ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ)
3713, 17, 19, 23, 35, 36syl32anc 1379 . . . . . . . . . . . 12 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ)
38 xrmax2 13155 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) β†’ 𝑦 ≀ if(0 ≀ 𝑦, 𝑦, 0))
3915, 14, 38sylancr 588 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑦 ≀ if(0 ≀ 𝑦, 𝑦, 0))
40 df-ioc 13329 . . . . . . . . . . . . . . 15 (,] = (π‘Ž ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (π‘Ž < 𝑐 ∧ 𝑐 ≀ 𝑏)})
41 xrlelttr 13135 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ* ∧ π‘₯ ∈ ℝ*) β†’ ((𝑦 ≀ if(0 ≀ 𝑦, 𝑦, 0) ∧ if(0 ≀ 𝑦, 𝑦, 0) < π‘₯) β†’ 𝑦 < π‘₯))
4240, 40, 41ixxss1 13342 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 𝑦 ≀ if(0 ≀ 𝑦, 𝑦, 0)) β†’ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† (𝑦(,]+∞))
4314, 39, 42syl2anc 585 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† (𝑦(,]+∞))
44 simplr 768 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑒 βŠ† 𝐴)
4525, 44eqsstrrd 4022 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (𝑦(,]+∞) βŠ† 𝐴)
4643, 45sstrd 3993 . . . . . . . . . . . 12 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† 𝐴)
47 oveq1 7416 . . . . . . . . . . . . . 14 (π‘₯ = if(0 ≀ 𝑦, 𝑦, 0) β†’ (π‘₯(,]+∞) = (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞))
4847sseq1d 4014 . . . . . . . . . . . . 13 (π‘₯ = if(0 ≀ 𝑦, 𝑦, 0) β†’ ((π‘₯(,]+∞) βŠ† 𝐴 ↔ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† 𝐴))
4948rspcev 3613 . . . . . . . . . . . 12 ((if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
5037, 46, 49syl2anc 585 . . . . . . . . . . 11 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
5150rexlimdvaa 3157 . . . . . . . . . 10 ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ (βˆƒπ‘¦ ∈ ℝ* 𝑒 = (𝑦(,]+∞) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
5251com12 32 . . . . . . . . 9 (βˆƒπ‘¦ ∈ ℝ* 𝑒 = (𝑦(,]+∞) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
5311, 52sylbi 216 . . . . . . . 8 (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
54 eqid 2733 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
5554elrnmpt 5956 . . . . . . . . . 10 (𝑒 ∈ V β†’ (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (-∞[,)𝑦)))
5655elv 3481 . . . . . . . . 9 (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (-∞[,)𝑦))
57 pnfnlt 13108 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* β†’ Β¬ +∞ < 𝑦)
58 elico1 13367 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) β†’ (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≀ +∞ ∧ +∞ < 𝑦)))
5912, 58mpan 689 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* β†’ (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≀ +∞ ∧ +∞ < 𝑦)))
60 simp3 1139 . . . . . . . . . . . . . . 15 ((+∞ ∈ ℝ* ∧ -∞ ≀ +∞ ∧ +∞ < 𝑦) β†’ +∞ < 𝑦)
6159, 60syl6bi 253 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* β†’ (+∞ ∈ (-∞[,)𝑦) β†’ +∞ < 𝑦))
6257, 61mtod 197 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* β†’ Β¬ +∞ ∈ (-∞[,)𝑦))
63 eleq2 2823 . . . . . . . . . . . . . 14 (𝑒 = (-∞[,)𝑦) β†’ (+∞ ∈ 𝑒 ↔ +∞ ∈ (-∞[,)𝑦)))
6463notbid 318 . . . . . . . . . . . . 13 (𝑒 = (-∞[,)𝑦) β†’ (Β¬ +∞ ∈ 𝑒 ↔ Β¬ +∞ ∈ (-∞[,)𝑦)))
6562, 64syl5ibrcom 246 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* β†’ (𝑒 = (-∞[,)𝑦) β†’ Β¬ +∞ ∈ 𝑒))
6665rexlimiv 3149 . . . . . . . . . . 11 (βˆƒπ‘¦ ∈ ℝ* 𝑒 = (-∞[,)𝑦) β†’ Β¬ +∞ ∈ 𝑒)
6766pm2.21d 121 . . . . . . . . . 10 (βˆƒπ‘¦ ∈ ℝ* 𝑒 = (-∞[,)𝑦) β†’ (+∞ ∈ 𝑒 β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
6867adantrd 493 . . . . . . . . 9 (βˆƒπ‘¦ ∈ ℝ* 𝑒 = (-∞[,)𝑦) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
6956, 68sylbi 216 . . . . . . . 8 (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
7053, 69jaoi 856 . . . . . . 7 ((𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
718, 70sylbi 216 . . . . . 6 (𝑒 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
72 pnfnre 11255 . . . . . . . . . 10 +∞ βˆ‰ ℝ
7372neli 3049 . . . . . . . . 9 Β¬ +∞ ∈ ℝ
74 elssuni 4942 . . . . . . . . . . 11 (𝑒 ∈ ran (,) β†’ 𝑒 βŠ† βˆͺ ran (,))
75 unirnioo 13426 . . . . . . . . . . 11 ℝ = βˆͺ ran (,)
7674, 75sseqtrrdi 4034 . . . . . . . . . 10 (𝑒 ∈ ran (,) β†’ 𝑒 βŠ† ℝ)
7776sseld 3982 . . . . . . . . 9 (𝑒 ∈ ran (,) β†’ (+∞ ∈ 𝑒 β†’ +∞ ∈ ℝ))
7873, 77mtoi 198 . . . . . . . 8 (𝑒 ∈ ran (,) β†’ Β¬ +∞ ∈ 𝑒)
7978pm2.21d 121 . . . . . . 7 (𝑒 ∈ ran (,) β†’ (+∞ ∈ 𝑒 β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
8079adantrd 493 . . . . . 6 (𝑒 ∈ ran (,) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
8171, 80jaoi 856 . . . . 5 ((𝑒 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ∨ 𝑒 ∈ ran (,)) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
827, 81sylbi 216 . . . 4 (𝑒 ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,)) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
8382rexlimiv 3149 . . 3 (βˆƒπ‘’ ∈ ((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,))(+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
846, 83syl 17 . 2 ((𝐴 ∈ (topGenβ€˜((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,))) ∧ +∞ ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
855, 84sylanb 582 1 ((𝐴 ∈ (ordTopβ€˜ ≀ ) ∧ +∞ ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  ifcif 4529  βˆͺ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  β„cr 11109  0cc0 11110  +∞cpnf 11245  -∞cmnf 11246  β„*cxr 11247   < clt 11248   ≀ cle 11249  (,)cioo 13324  (,]cioc 13325  [,)cico 13326  topGenctg 17383  ordTopcordt 17445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-ioo 13328  df-ioc 13329  df-ico 13330  df-icc 13331  df-topgen 17389  df-ordt 17447  df-ps 18519  df-tsr 18520  df-top 22396  df-bases 22449
This theorem is referenced by:  xrge0tsms  24350  xrlimcnp  26473  xrge0tsmsd  32240  pnfneige0  32962  xlimpnfvlem2  44601
  Copyright terms: Public domain W3C validator