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Mirrors > Home > MPE Home > Th. List > supxrre1 | Structured version Visualization version GIF version |
Description: The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.) |
Ref | Expression |
---|---|
supxrre1 | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) < +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supxrgtmnf 13292 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ*, < )) | |
2 | ressxr 11242 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
3 | sstr 3987 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ*) → 𝐴 ⊆ ℝ*) | |
4 | 2, 3 | mpan2 689 | . . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ*) |
5 | supxrcl 13278 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
6 | xrrebnd 13131 | . . . 4 ⊢ (sup(𝐴, ℝ*, < ) ∈ ℝ* → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ*, < ) ∧ sup(𝐴, ℝ*, < ) < +∞))) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 ⊢ (𝐴 ⊆ ℝ → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ*, < ) ∧ sup(𝐴, ℝ*, < ) < +∞))) |
8 | 7 | adantr 481 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ*, < ) ∧ sup(𝐴, ℝ*, < ) < +∞))) |
9 | 1, 8 | mpbirand 705 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3945 ∅c0 4319 class class class wbr 5142 supcsup 9419 ℝcr 11093 +∞cpnf 11229 -∞cmnf 11230 ℝ*cxr 11231 < clt 11232 |
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 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-pre-sup 11172 |
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-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-po 5582 df-so 5583 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-sup 9421 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 |
This theorem is referenced by: supxrre2 13294 limsupgre 15409 supxrre3 43872 |
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