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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 3156 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
4 | lttri3 8014 | . . . 4 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) | |
5 | 4 | adantl 277 | . . 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 6990 | . 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
8 | 5, 6 | supubti 6991 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵)) |
9 | 2, 8 | mpd 13 | . 2 ⊢ (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵) |
10 | 3, 7, 9 | nltled 8055 | 1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < )) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ⊆ wss 3129 class class class wbr 4000 supcsup 6974 ℝcr 7788 < clt 7969 ≤ cle 7970 |
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 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-pre-ltirr 7901 ax-pre-apti 7904 |
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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4628 df-cnv 4630 df-iota 5173 df-riota 5824 df-sup 6976 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 |
This theorem is referenced by: suprzclex 9327 |
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