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Mirrors > Home > ILE Home > Th. List > inflbti | GIF version |
Description: An infimum is a lower bound. See also infclti 6436 and infglbti 6438. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infclti.ti | ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) |
infclti.ex | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
Ref | Expression |
---|---|
inflbti | ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infclti.ti | . . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) | |
2 | 1 | cnvti 6432 | . . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢))) |
3 | infclti.ex | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
4 | 3 | cnvinfex 6431 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
5 | 2, 4 | supubti 6412 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
6 | 5 | imp 122 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶) |
7 | df-inf 6398 | . . . . . 6 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
8 | 7 | a1i 9 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)) |
9 | 8 | breq2d 3797 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅))) |
10 | 2, 4 | supclti 6411 | . . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
11 | brcnvg 4534 | . . . . . 6 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶 ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅))) | |
12 | 11 | bicomd 139 | . . . . 5 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
13 | 10, 12 | sylan 277 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
14 | 9, 13 | bitrd 186 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
15 | 6, 14 | mtbird 630 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)) |
16 | 15 | ex 113 | 1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 class class class wbr 3785 ◡ccnv 4362 supcsup 6395 infcinf 6396 |
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-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 |
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-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 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-cnv 4371 df-iota 4887 df-riota 5488 df-sup 6397 df-inf 6398 |
This theorem is referenced by: zssinfcl 10344 infssuzledc 10346 |
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