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Theorem pnfnei 22715
Description: A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24313 (which describes neighborhoods of ℝ) and mnfnei 22716, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 22712 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 2732 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2732 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
3 eqid 2732 . . . 4 ran (,) = ran (,)
41, 2, 3leordtval 22708 . . 3 (ordTopβ€˜ ≀ ) = (topGenβ€˜((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,)))
54eleq2i 2825 . 2 (𝐴 ∈ (ordTopβ€˜ ≀ ) ↔ 𝐴 ∈ (topGenβ€˜((ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) βˆͺ ran (,))))
6 tg2 22459 . . 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 4147 . . . . 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 4147 . . . . . . 7 (𝑒 ∈ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) βˆͺ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ↔ (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∨ 𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))
9 eqid 2732 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
109elrnmpt 5953 . . . . . . . . . 10 (𝑒 ∈ V β†’ (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (𝑦(,]+∞)))
1110elv 3480 . . . . . . . . 9 (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (𝑦(,]+∞))
12 mnfxr 11267 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
1312a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ -∞ ∈ ℝ*)
14 simprl 769 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑦 ∈ ℝ*)
15 0xr 11257 . . . . . . . . . . . . . 14 0 ∈ ℝ*
16 ifcl 4572 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ*)
1714, 15, 16sylancl 586 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ*)
18 pnfxr 11264 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
1918a1i 11 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ +∞ ∈ ℝ*)
20 xrmax1 13150 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) β†’ 0 ≀ if(0 ≀ 𝑦, 𝑦, 0))
2115, 14, 20sylancr 587 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 0 ≀ if(0 ≀ 𝑦, 𝑦, 0))
22 ge0gtmnf 13147 . . . . . . . . . . . . . 14 ((if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ* ∧ 0 ≀ if(0 ≀ 𝑦, 𝑦, 0)) β†’ -∞ < if(0 ≀ 𝑦, 𝑦, 0))
2317, 21, 22syl2anc 584 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ -∞ < if(0 ≀ 𝑦, 𝑦, 0))
24 simpll 765 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ +∞ ∈ 𝑒)
25 simprr 771 . . . . . . . . . . . . . . . . 17 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑒 = (𝑦(,]+∞))
2624, 25eleqtrd 2835 . . . . . . . . . . . . . . . 16 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ +∞ ∈ (𝑦(,]+∞))
27 elioc1 13362 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) β†’ (+∞ ∈ (𝑦(,]+∞) ↔ (+∞ ∈ ℝ* ∧ 𝑦 < +∞ ∧ +∞ ≀ +∞)))
2814, 18, 27sylancl 586 . . . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑦 < +∞)
31 0ltpnf 13098 . . . . . . . . . . . . . 14 0 < +∞
32 breq1 5150 . . . . . . . . . . . . . . 15 (𝑦 = if(0 ≀ 𝑦, 𝑦, 0) β†’ (𝑦 < +∞ ↔ if(0 ≀ 𝑦, 𝑦, 0) < +∞))
33 breq1 5150 . . . . . . . . . . . . . . 15 (0 = if(0 ≀ 𝑦, 𝑦, 0) β†’ (0 < +∞ ↔ if(0 ≀ 𝑦, 𝑦, 0) < +∞))
3432, 33ifboth 4566 . . . . . . . . . . . . . 14 ((𝑦 < +∞ ∧ 0 < +∞) β†’ if(0 ≀ 𝑦, 𝑦, 0) < +∞)
3530, 31, 34sylancl 586 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ if(0 ≀ 𝑦, 𝑦, 0) < +∞)
36 xrre2 13145 . . . . . . . . . . . . 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 1378 . . . . . . . . . . . 12 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ)
38 xrmax2 13151 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) β†’ 𝑦 ≀ if(0 ≀ 𝑦, 𝑦, 0))
3915, 14, 38sylancr 587 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑦 ≀ if(0 ≀ 𝑦, 𝑦, 0))
40 df-ioc 13325 . . . . . . . . . . . . . . 15 (,] = (π‘Ž ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (π‘Ž < 𝑐 ∧ 𝑐 ≀ 𝑏)})
41 xrlelttr 13131 . . . . . . . . . . . . . . 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 13338 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ 𝑦 ≀ if(0 ≀ 𝑦, 𝑦, 0)) β†’ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† (𝑦(,]+∞))
4314, 39, 42syl2anc 584 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† (𝑦(,]+∞))
44 simplr 767 . . . . . . . . . . . . . 14 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ 𝑒 βŠ† 𝐴)
4525, 44eqsstrrd 4020 . . . . . . . . . . . . 13 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (𝑦(,]+∞) βŠ† 𝐴)
4643, 45sstrd 3991 . . . . . . . . . . . 12 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† 𝐴)
47 oveq1 7412 . . . . . . . . . . . . . 14 (π‘₯ = if(0 ≀ 𝑦, 𝑦, 0) β†’ (π‘₯(,]+∞) = (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞))
4847sseq1d 4012 . . . . . . . . . . . . 13 (π‘₯ = if(0 ≀ 𝑦, 𝑦, 0) β†’ ((π‘₯(,]+∞) βŠ† 𝐴 ↔ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† 𝐴))
4948rspcev 3612 . . . . . . . . . . . 12 ((if(0 ≀ 𝑦, 𝑦, 0) ∈ ℝ ∧ (if(0 ≀ 𝑦, 𝑦, 0)(,]+∞) βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
5037, 46, 49syl2anc 584 . . . . . . . . . . 11 (((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) ∧ (𝑦 ∈ ℝ* ∧ 𝑒 = (𝑦(,]+∞))) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴)
5150rexlimdvaa 3156 . . . . . . . . . 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 2732 . . . . . . . . . . 11 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
5554elrnmpt 5953 . . . . . . . . . 10 (𝑒 ∈ V β†’ (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (-∞[,)𝑦)))
5655elv 3480 . . . . . . . . 9 (𝑒 ∈ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ↔ βˆƒπ‘¦ ∈ ℝ* 𝑒 = (-∞[,)𝑦))
57 pnfnlt 13104 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* β†’ Β¬ +∞ < 𝑦)
58 elico1 13363 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) β†’ (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≀ +∞ ∧ +∞ < 𝑦)))
5912, 58mpan 688 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* β†’ (+∞ ∈ (-∞[,)𝑦) ↔ (+∞ ∈ ℝ* ∧ -∞ ≀ +∞ ∧ +∞ < 𝑦)))
60 simp3 1138 . . . . . . . . . . . . . . 15 ((+∞ ∈ ℝ* ∧ -∞ ≀ +∞ ∧ +∞ < 𝑦) β†’ +∞ < 𝑦)
6159, 60syl6bi 252 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* β†’ (+∞ ∈ (-∞[,)𝑦) β†’ +∞ < 𝑦))
6257, 61mtod 197 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* β†’ Β¬ +∞ ∈ (-∞[,)𝑦))
63 eleq2 2822 . . . . . . . . . . . . . 14 (𝑒 = (-∞[,)𝑦) β†’ (+∞ ∈ 𝑒 ↔ +∞ ∈ (-∞[,)𝑦)))
6463notbid 317 . . . . . . . . . . . . 13 (𝑒 = (-∞[,)𝑦) β†’ (Β¬ +∞ ∈ 𝑒 ↔ Β¬ +∞ ∈ (-∞[,)𝑦)))
6562, 64syl5ibrcom 246 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* β†’ (𝑒 = (-∞[,)𝑦) β†’ Β¬ +∞ ∈ 𝑒))
6665rexlimiv 3148 . . . . . . . . . . 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 492 . . . . . . . . 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 855 . . . . . . 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 11251 . . . . . . . . . 10 +∞ βˆ‰ ℝ
7372neli 3048 . . . . . . . . 9 Β¬ +∞ ∈ ℝ
74 elssuni 4940 . . . . . . . . . . 11 (𝑒 ∈ ran (,) β†’ 𝑒 βŠ† βˆͺ ran (,))
75 unirnioo 13422 . . . . . . . . . . 11 ℝ = βˆͺ ran (,)
7674, 75sseqtrrdi 4032 . . . . . . . . . 10 (𝑒 ∈ ran (,) β†’ 𝑒 βŠ† ℝ)
7776sseld 3980 . . . . . . . . 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 492 . . . . . 6 (𝑒 ∈ ran (,) β†’ ((+∞ ∈ 𝑒 ∧ 𝑒 βŠ† 𝐴) β†’ βˆƒπ‘₯ ∈ ℝ (π‘₯(,]+∞) βŠ† 𝐴))
8171, 80jaoi 855 . . . . 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 3148 . . 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 581 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474   βˆͺ cun 3945   βŠ† wss 3947  ifcif 4527  βˆͺ cuni 4907   class class class wbr 5147   ↦ cmpt 5230  ran crn 5676  β€˜cfv 6540  (class class class)co 7405  β„cr 11105  0cc0 11106  +∞cpnf 11241  -∞cmnf 11242  β„*cxr 11243   < clt 11244   ≀ cle 11245  (,)cioo 13320  (,]cioc 13321  [,)cico 13322  topGenctg 17379  ordTopcordt 17441
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-ioo 13324  df-ioc 13325  df-ico 13326  df-icc 13327  df-topgen 17385  df-ordt 17443  df-ps 18515  df-tsr 18516  df-top 22387  df-bases 22440
This theorem is referenced by:  xrge0tsms  24341  xrlimcnp  26462  xrge0tsmsd  32196  pnfneige0  32919  xlimpnfvlem2  44539
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