| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > ILE Home > Th. List > suprleubex | GIF version | ||
| Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.) |
| 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 |
|---|---|
| suprleubex | ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lttri3 8152 | . . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) | |
| 2 | 1 | adantl 277 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
| 3 | suprubex.ex | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
| 4 | 2, 3 | supclti 7100 | . . . . . 6 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
| 5 | suprlubex.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 6 | 4, 5 | lenltd 8190 | . . . . 5 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup(𝐴, ℝ, < ))) |
| 7 | suprubex.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
| 8 | 3, 7, 5 | suprnubex 9026 | . . . . 5 ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) |
| 9 | 6, 8 | bitrd 188 | . . . 4 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧)) |
| 10 | breq2 4048 | . . . . . 6 ⊢ (𝑤 = 𝑧 → (𝐵 < 𝑤 ↔ 𝐵 < 𝑧)) | |
| 11 | 10 | notbid 669 | . . . . 5 ⊢ (𝑤 = 𝑧 → (¬ 𝐵 < 𝑤 ↔ ¬ 𝐵 < 𝑧)) |
| 12 | 11 | cbvralv 2738 | . . . 4 ⊢ (∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧) |
| 13 | 9, 12 | bitr4di 198 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤)) |
| 14 | 7 | sselda 3193 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ) |
| 15 | 5 | adantr 276 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝐵 ∈ ℝ) |
| 16 | 14, 15 | lenltd 8190 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤)) |
| 17 | 16 | ralbidva 2502 | . . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤)) |
| 18 | 13, 17 | bitr4d 191 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵)) |
| 19 | breq1 4047 | . . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝐵 ↔ 𝑧 ≤ 𝐵)) | |
| 20 | 19 | cbvralv 2738 | . 2 ⊢ (∀𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵) |
| 21 | 18, 20 | bitrdi 196 | 1 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)) |
| Colors of variables: wff set class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2176 ∀wral 2484 ∃wrex 2485 ⊆ wss 3166 class class class wbr 4044 supcsup 7084 ℝcr 7924 < clt 8107 ≤ cle 8108 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-po 4343 df-iso 4344 df-xp 4681 df-cnv 4683 df-iota 5232 df-riota 5899 df-sup 7086 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 |
| This theorem is referenced by: suprzclex 9471 suplociccex 15097 |
| Copyright terms: Public domain | W3C validator |