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Mirrors > Home > ILE Home > Th. List > suprlubex | GIF version |
Description: The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.) |
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 |
---|---|
suprlubex | ⊢ (𝜑 → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprlubex.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | lttri3 7844 | . . . 4 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) | |
3 | 2 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
4 | suprubex.ex | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
5 | ltso 7842 | . . . 4 ⊢ < Or ℝ | |
6 | 5 | a1i 9 | . . 3 ⊢ (𝜑 → < Or ℝ) |
7 | suprubex.ss | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
8 | 3, 4, 6, 7 | suplub2ti 6888 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)) |
9 | 1, 8 | mpdan 417 | 1 ⊢ (𝜑 → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ⊆ wss 3071 class class class wbr 3929 Or wor 4217 supcsup 6869 ℝcr 7619 < clt 7800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-po 4218 df-iso 4219 df-xp 4545 df-iota 5088 df-riota 5730 df-sup 6871 df-pnf 7802 df-mnf 7803 df-ltxr 7805 |
This theorem is referenced by: suprnubex 8711 suprzclex 9149 |
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