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Mirrors > Home > MPE Home > Th. List > suprub | Structured version Visualization version GIF version |
Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.) |
Ref | Expression |
---|---|
suprub | ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → 𝐴 ⊆ ℝ) | |
2 | 1 | sselda 3921 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ℝ) |
3 | suprcl 11935 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
4 | 3 | adantr 481 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ ℝ) |
5 | ltso 11055 | . . . . 5 ⊢ < Or ℝ | |
6 | 5 | a1i 11 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → < Or ℝ) |
7 | sup3 11932 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
8 | 6, 7 | supub 9218 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵)) |
9 | 8 | imp 407 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ 𝐴) → ¬ sup(𝐴, ℝ, < ) < 𝐵) |
10 | 2, 4, 9 | nltled 11125 | 1 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 Or wor 5502 supcsup 9199 ℝcr 10870 < clt 11009 ≤ cle 11010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 |
This theorem is referenced by: suprubd 11937 supaddc 11942 supadd 11943 supmul1 11944 supmullem1 11945 supmul 11947 suprubii 11950 suprzcl 12400 rpnnen1lem5 12721 supicc 13233 sqrlem4 14957 sqrlem7 14960 isercolllem2 15377 climsup 15381 fsumcvg3 15441 supcvg 15568 mertenslem1 15596 evth 24122 ivthlem2 24616 ivthlem3 24617 itg2mono 24918 esumpcvgval 32046 erdszelem8 33160 itg2addnclem2 35829 ftc1anclem7 35856 ftc1anc 35858 totbndbnd 35947 prdsbnd 35951 ubelsupr 42563 suprnmpt 42710 upbdrech 42844 ssfiunibd 42848 uzfissfz 42865 fourierdlem20 43668 fourierdlem31 43679 fourierdlem64 43711 fourierdlem79 43726 sge0isum 43965 hoicvr 44086 hoidmv1lelem1 44129 hoidmv1lelem3 44131 hoidmvlelem1 44133 hoidmvlelem4 44136 |
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