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Mirrors > Home > MPE Home > Th. List > supxrpnf | Structured version Visualization version GIF version |
Description: The supremum of a set of extended reals containing plus infinity is plus infinity. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
supxrpnf | ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3961 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ*)) | |
2 | pnfnlt 12517 | . . . . 5 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦) | |
3 | 1, 2 | syl6 35 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦)) |
4 | 3 | ralrimiv 3181 | . . 3 ⊢ (𝐴 ⊆ ℝ* → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦) |
5 | breq2 5063 | . . . . . 6 ⊢ (𝑧 = +∞ → (𝑦 < 𝑧 ↔ 𝑦 < +∞)) | |
6 | 5 | rspcev 3623 | . . . . 5 ⊢ ((+∞ ∈ 𝐴 ∧ 𝑦 < +∞) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) |
7 | 6 | ex 415 | . . . 4 ⊢ (+∞ ∈ 𝐴 → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
8 | 7 | ralrimivw 3183 | . . 3 ⊢ (+∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
9 | 4, 8 | anim12i 614 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
10 | pnfxr 10689 | . . 3 ⊢ +∞ ∈ ℝ* | |
11 | supxr 12700 | . . 3 ⊢ (((𝐴 ⊆ ℝ* ∧ +∞ ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ*, < ) = +∞) | |
12 | 10, 11 | mpanl2 699 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → sup(𝐴, ℝ*, < ) = +∞) |
13 | 9, 12 | syldan 593 | 1 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5059 supcsup 8898 ℝcr 10530 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 |
This theorem is referenced by: xrsup 13230 volsup 24151 supxrge 41598 supminfxr2 41737 sge0tsms 42655 sge0sup 42666 |
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