Mathbox for Stanislas Polu |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > suprleubrd | Structured version Visualization version GIF version |
Description: Natural deduction form of specialized suprleub 11607. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
suprleubrd.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
suprleubrd.2 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
suprleubrd.3 | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
suprleubrd.4 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
suprleubrd.5 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) |
Ref | Expression |
---|---|
suprleubrd | ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprleubrd.5 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) | |
2 | suprleubrd.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
3 | suprleubrd.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
4 | suprleubrd.3 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | |
5 | suprleubrd.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | suprleub 11607 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) | |
7 | 2, 3, 4, 5, 6 | syl31anc 1369 | . . . . 5 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) |
8 | 7 | bicomd 225 | . . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 ↔ sup(𝐴, ℝ, < ) ≤ 𝐵)) |
9 | 8 | biimpd 231 | . . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 → sup(𝐴, ℝ, < ) ≤ 𝐵)) |
10 | 9 | imp 409 | . 2 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) → sup(𝐴, ℝ, < ) ≤ 𝐵) |
11 | 1, 10 | mpdan 685 | 1 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 supcsup 8904 ℝcr 10536 < clt 10675 ≤ cle 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 |
This theorem is referenced by: imo72b2lem2 40538 |
Copyright terms: Public domain | W3C validator |