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Mirrors > Home > ILE Home > Th. List > suprnubex | GIF version |
Description: An upper bound is not less than the supremum of a nonempty bounded set of reals. (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 |
---|---|
suprnubex | ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprubex.ex | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
2 | suprubex.ss | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
3 | suprlubex.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 1, 2, 3 | suprlubex 8908 | . . 3 ⊢ (𝜑 → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)) |
5 | 4 | notbid 667 | . 2 ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ¬ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)) |
6 | ralnex 2465 | . 2 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧 ↔ ¬ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧) | |
7 | 5, 6 | bitr4di 198 | 1 ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) |
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 4003 supcsup 6980 ℝcr 7809 < clt 7991 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-po 4296 df-iso 4297 df-xp 4632 df-iota 5178 df-riota 5830 df-sup 6982 df-pnf 7993 df-mnf 7994 df-ltxr 7996 |
This theorem is referenced by: suprleubex 8910 |
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