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Mirrors > Home > ILE Home > Th. List > lble | GIF version |
Description: If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
lble | ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbreu 8905 | . . . . 5 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) | |
2 | nfcv 2319 | . . . . . . 7 ⊢ Ⅎ𝑥𝑆 | |
3 | nfriota1 5841 | . . . . . . . 8 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) | |
4 | nfcv 2319 | . . . . . . . 8 ⊢ Ⅎ𝑥 ≤ | |
5 | nfcv 2319 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
6 | 3, 4, 5 | nfbr 4051 | . . . . . . 7 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 |
7 | 2, 6 | nfralxy 2515 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 |
8 | eqid 2177 | . . . . . 6 ⊢ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) | |
9 | nfra1 2508 | . . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 | |
10 | nfcv 2319 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝑆 | |
11 | 9, 10 | nfriota 5843 | . . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) |
12 | 11 | nfeq2 2331 | . . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) |
13 | breq1 4008 | . . . . . . 7 ⊢ (𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (𝑥 ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦)) | |
14 | 12, 13 | ralbid 2475 | . . . . . 6 ⊢ (𝑥 = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦)) |
15 | 7, 8, 14 | riotaprop 5857 | . . . . 5 ⊢ (∃!𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦)) |
16 | 1, 15 | syl 14 | . . . 4 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦)) |
17 | 16 | simprd 114 | . . 3 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦) |
18 | nfcv 2319 | . . . . 5 ⊢ Ⅎ𝑦 ≤ | |
19 | nfcv 2319 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
20 | 11, 18, 19 | nfbr 4051 | . . . 4 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴 |
21 | breq2 4009 | . . . 4 ⊢ (𝑦 = 𝐴 → ((℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 ↔ (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)) | |
22 | 20, 21 | rspc 2837 | . . 3 ⊢ (𝐴 ∈ 𝑆 → (∀𝑦 ∈ 𝑆 (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝑦 → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴)) |
23 | 17, 22 | mpan9 281 | . 2 ⊢ (((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴) |
24 | 23 | 3impa 1194 | 1 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ∃!wreu 2457 ⊆ wss 3131 class class class wbr 4005 ℩crio 5833 ℝcr 7813 ≤ cle 7996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-pre-ltirr 7926 ax-pre-apti 7929 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-iota 5180 df-riota 5834 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 |
This theorem is referenced by: lbinf 8908 lbinfle 8910 |
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