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Mirrors > Home > MPE Home > Th. List > infempty | Structured version Visualization version GIF version |
Description: The infimum of an empty set under a base set which has a unique greatest element is the greatest element of the base set. (Contributed by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
infempty | ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 8901 | . 2 ⊢ inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, ◡𝑅) | |
2 | cnvso 6133 | . . 3 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
3 | brcnvg 5744 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦)) | |
4 | 3 | ancoms 461 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦)) |
5 | 4 | bicomd 225 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 ↔ 𝑦◡𝑅𝑋)) |
6 | 5 | notbid 320 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑋)) |
7 | 6 | ralbidva 3196 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋)) |
8 | 7 | pm5.32i 577 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋)) |
9 | brcnvg 5744 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) | |
10 | 9 | ancoms 461 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
11 | 10 | bicomd 225 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥)) |
12 | 11 | notbid 320 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥)) |
13 | 12 | ralbidva 3196 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥)) |
14 | 13 | reubiia 3390 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥) |
15 | sup0 8924 | . . 3 ⊢ ((◡𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥) → sup(∅, 𝐴, ◡𝑅) = 𝑋) | |
16 | 2, 8, 14, 15 | syl3anb 1157 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, ◡𝑅) = 𝑋) |
17 | 1, 16 | syl5eq 2868 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃!wreu 3140 ∅c0 4290 class class class wbr 5058 Or wor 5467 ◡ccnv 5548 supcsup 8898 infcinf 8899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-po 5468 df-so 5469 df-cnv 5557 df-iota 6308 df-riota 7108 df-sup 8900 df-inf 8901 |
This theorem is referenced by: (None) |
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