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Mirrors > Home > ILE Home > Th. List > suprleubex | 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 |
---|---|
suprubex.ex | ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
suprubex.ss | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
suprlubex.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
suprleubex | ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7662 | . . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) | |
2 | 1 | adantl 272 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
3 | suprubex.ex | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
4 | 2, 3 | supclti 6773 | . . . . . 6 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
5 | suprlubex.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 4, 5 | lenltd 7698 | . . . . 5 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < ))) |
7 | suprubex.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
8 | 3, 7, 5 | suprnubex 8511 | . . . . 5 ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) |
9 | 6, 8 | bitrd 187 | . . . 4 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) |
10 | breq2 3871 | . . . . . 6 ⊢ (𝑤 = 𝑧 → (𝐵 < 𝑤 ↔ 𝐵 < 𝑧)) | |
11 | 10 | notbid 630 | . . . . 5 ⊢ (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧)) |
12 | 11 | cbvralv 2604 | . . . 4 ⊢ (∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧) |
13 | 9, 12 | syl6bbr 197 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤)) |
14 | 7 | sselda 3039 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ) |
15 | 5 | adantr 271 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝐵 ∈ ℝ) |
16 | 14, 15 | lenltd 7698 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤)) |
17 | 16 | ralbidva 2387 | . . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤)) |
18 | 13, 17 | bitr4d 190 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵)) |
19 | breq1 3870 | . . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝐵 ↔ 𝑧 ≤ 𝐵)) | |
20 | 19 | cbvralv 2604 | . 2 ⊢ (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) |
21 | 18, 20 | syl6bb 195 | 1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1445 ∀wral 2370 ∃wrex 2371 ⊆ wss 3013 class class class wbr 3867 supcsup 6757 ℝcr 7446 < clt 7619 ≤ cle 7620 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-po 4147 df-iso 4148 df-xp 4473 df-cnv 4475 df-iota 5014 df-riota 5646 df-sup 6759 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 |
This theorem is referenced by: suprzclex 8943 |
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