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Mirrors > Home > MPE Home > Th. List > supxrgtmnf | Structured version Visualization version GIF version |
Description: The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
supxrgtmnf | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supxrbnd 12806 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ*, < ) < +∞) → sup(𝐴, ℝ*, < ) ∈ ℝ) | |
2 | 1 | 3expia 1122 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) < +∞ → sup(𝐴, ℝ*, < ) ∈ ℝ)) |
3 | 2 | con3d 155 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (¬ sup(𝐴, ℝ*, < ) ∈ ℝ → ¬ sup(𝐴, ℝ*, < ) < +∞)) |
4 | ressxr 10765 | . . . . . . . 8 ⊢ ℝ ⊆ ℝ* | |
5 | sstr 3885 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ*) → 𝐴 ⊆ ℝ*) | |
6 | 4, 5 | mpan2 691 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ*) |
7 | supxrcl 12793 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝐴 ⊆ ℝ → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | nltpnft 12642 | . . . . 5 ⊢ (sup(𝐴, ℝ*, < ) ∈ ℝ* → (sup(𝐴, ℝ*, < ) = +∞ ↔ ¬ sup(𝐴, ℝ*, < ) < +∞)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) = +∞ ↔ ¬ sup(𝐴, ℝ*, < ) < +∞)) |
12 | 3, 11 | sylibrd 262 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (¬ sup(𝐴, ℝ*, < ) ∈ ℝ → sup(𝐴, ℝ*, < ) = +∞)) |
13 | 12 | orrd 862 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ∨ sup(𝐴, ℝ*, < ) = +∞)) |
14 | mnfltxr 12607 | . 2 ⊢ ((sup(𝐴, ℝ*, < ) ∈ ℝ ∨ sup(𝐴, ℝ*, < ) = +∞) → -∞ < sup(𝐴, ℝ*, < )) | |
15 | 13, 14 | syl 17 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ⊆ wss 3843 ∅c0 4211 class class class wbr 5030 supcsup 8979 ℝcr 10616 +∞cpnf 10752 -∞cmnf 10753 ℝ*cxr 10754 < clt 10755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-sup 8981 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 |
This theorem is referenced by: supxrre1 12808 ovolunlem1a 24250 suplesup 42438 |
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