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Mirrors > Home > ILE Home > Th. List > suprubex | GIF version |
Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-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) |
suprubex.b | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
suprubex | ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprubex.ss | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | suprubex.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
3 | 1, 2 | sseldd 3000 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
4 | lttri3 7191 | . . . 4 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) | |
5 | 4 | adantl 271 | . . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
6 | suprubex.ex | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
7 | 5, 6 | supclti 6411 | . 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
8 | 5, 6 | supubti 6412 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵)) |
9 | 2, 8 | mpd 13 | . 2 ⊢ (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵) |
10 | 3, 7, 9 | nltled 7230 | 1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < )) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 ⊆ wss 2973 class class class wbr 3785 supcsup 6395 ℝcr 6980 < clt 7153 ≤ cle 7154 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-apti 7091 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-iota 4887 df-riota 5488 df-sup 6397 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 |
This theorem is referenced by: suprzclex 8445 |
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