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Mirrors > Home > MPE Home > Th. List > inflb | Structured version Visualization version GIF version |
Description: An infimum is a lower bound. See also infcl 9511 and infglb 9513. (Contributed by AV, 3-Sep-2020.) |
Ref | Expression |
---|---|
infcl.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
infcl.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
Ref | Expression |
---|---|
inflb | ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infcl.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
2 | cnvso 6292 | . . . . . 6 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
3 | 1, 2 | sylib 217 | . . . . 5 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
4 | infcl.2 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
5 | 1, 4 | infcllem 9510 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
6 | 3, 5 | supub 9482 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
7 | 6 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶) |
8 | df-inf 9466 | . . . . . 6 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)) |
10 | 9 | breq2d 5160 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅))) |
11 | 3, 5 | supcl 9481 | . . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
12 | brcnvg 5882 | . . . . . 6 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶 ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅))) | |
13 | 12 | bicomd 222 | . . . . 5 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
14 | 11, 13 | sylan 579 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
15 | 10, 14 | bitrd 279 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)) |
16 | 7, 15 | mtbird 325 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)) |
17 | 16 | ex 412 | 1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 class class class wbr 5148 Or wor 5589 ◡ccnv 5677 supcsup 9463 infcinf 9464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-po 5590 df-so 5591 df-cnv 5686 df-iota 6500 df-riota 7376 df-sup 9465 df-inf 9466 |
This theorem is referenced by: infrelb 12229 infxrlb 13345 infssd 32492 infxrge0lb 32534 omssubadd 33920 ballotlemimin 34125 ballotlemfrcn0 34149 wsuclb 35424 |
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