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Mirrors > Home > MPE Home > Th. List > suprleub | Structured version Visualization version GIF version |
Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.) |
Ref | Expression |
---|---|
suprleub | ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprnub 12217 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤)) | |
2 | suprcl 12212 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
3 | lenlt 11330 | . . . 4 ⊢ ((sup(𝐴, ℝ, < ) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < ))) | |
4 | 2, 3 | sylan 578 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < ))) |
5 | simpl1 1188 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → 𝐴 ⊆ ℝ) | |
6 | 5 | sselda 3982 | . . . . 5 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ) |
7 | simplr 767 | . . . . 5 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) ∧ 𝑤 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
8 | 6, 7 | lenltd 11398 | . . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤)) |
9 | 8 | ralbidva 3173 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤)) |
10 | 1, 4, 9 | 3bitr4d 310 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵)) |
11 | breq1 5155 | . . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝐵 ↔ 𝑧 ≤ 𝐵)) | |
12 | 11 | cbvralvw 3232 | . 2 ⊢ (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) |
13 | 10, 12 | bitrdi 286 | 1 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 supcsup 9471 ℝcr 11145 < clt 11286 ≤ cle 11287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 |
This theorem is referenced by: supaddc 12219 supadd 12220 supmul1 12221 supmul 12224 suprleubii 12230 suprzcl 12680 rpnnen1lem3 13001 rpnnen1lem5 13003 supxrre 13346 supicc 13518 flval3 13820 01sqrexlem4 15232 01sqrexlem6 15234 ruclem12 16225 icccmplem3 24760 reconnlem2 24763 evth 24905 ivthlem2 25401 ivthlem3 25402 mbflimsup 25615 itg2cnlem1 25711 plyeq0lem 26164 ismblfin 37167 suprleubrd 43627 ubelsupr 44413 suprleubrnmpt 44833 sge0isum 45844 hoidmv1lelem1 46008 hoidmvlelem1 46012 |
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