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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 3171 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
4 | lttri3 8068 | . . . 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 7028 | . 2 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
8 | 5, 6 | supubti 7029 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ, < ) < 𝐵)) |
9 | 2, 8 | mpd 13 | . 2 ⊢ (𝜑 → ¬ sup(𝐴, ℝ, < ) < 𝐵) |
10 | 3, 7, 9 | nltled 8109 | 1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < )) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 ⊆ wss 3144 class class class wbr 4018 supcsup 7012 ℝcr 7841 < clt 8023 ≤ cle 8024 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-pre-ltirr 7954 ax-pre-apti 7957 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-iota 5196 df-riota 5852 df-sup 7014 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 |
This theorem is referenced by: suprzclex 9382 |
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