Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > supxrre3 | Structured version Visualization version GIF version |
Description: The supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
supxrre3 | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supxrre1 13106 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) < +∞)) | |
2 | id 22 | . . . . . 6 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ) | |
3 | rexr 11063 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
4 | 3 | ssriv 3930 | . . . . . . 7 ⊢ ℝ ⊆ ℝ* |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ⊆ ℝ → ℝ ⊆ ℝ*) |
6 | 2, 5 | sstrd 3936 | . . . . 5 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ*) |
7 | supxrbnd2 13098 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞)) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞)) |
9 | 8 | bicomd 222 | . . 3 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ*, < ) < +∞ ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
10 | 9 | adantr 482 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) < +∞ ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
11 | 1, 10 | bitrd 280 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2104 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ⊆ wss 3892 ∅c0 4262 class class class wbr 5081 supcsup 9239 ℝcr 10912 +∞cpnf 11048 ℝ*cxr 11050 < clt 11051 ≤ cle 11052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5496 df-po 5510 df-so 5511 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-sup 9241 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 |
This theorem is referenced by: supxrre3rnmpt 43016 sge0resplit 43993 |
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