![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > infmrgelbi | Structured version Visualization version GIF version |
Description: Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.) (Revised by AV, 17-Sep-2020.) |
Ref | Expression |
---|---|
infmrgelbi | ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → 𝐵 ≤ inf(𝐴, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) | |
2 | simpl1 1190 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → 𝐴 ⊆ ℝ) | |
3 | simpl2 1191 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → 𝐴 ≠ ∅) | |
4 | breq1 5150 | . . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑥 ↔ 𝐵 ≤ 𝑥)) | |
5 | 4 | ralbidv 3175 | . . . . 5 ⊢ (𝑧 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) |
6 | 5 | rspcev 3621 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑧 ≤ 𝑥) |
7 | 6 | 3ad2antl3 1186 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑧 ≤ 𝑥) |
8 | simpl3 1192 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → 𝐵 ∈ ℝ) | |
9 | infregelb 12249 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑧 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) | |
10 | 2, 3, 7, 8, 9 | syl31anc 1372 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) |
11 | 1, 10 | mpbird 257 | 1 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → 𝐵 ≤ inf(𝐴, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 infcinf 9478 ℝcr 11151 < clt 11292 ≤ cle 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 |
This theorem is referenced by: pellfundge 42869 pellfundglb 42872 |
Copyright terms: Public domain | W3C validator |