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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 8349 | . 2 ⊢ inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, ◡𝑅) | |
2 | cnvso 5674 | . . 3 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
3 | brcnvg 5303 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦)) | |
4 | 3 | ancoms 469 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦)) |
5 | 4 | bicomd 213 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 ↔ 𝑦◡𝑅𝑋)) |
6 | 5 | notbid 308 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑋)) |
7 | 6 | ralbidva 2985 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋)) |
8 | 7 | pm5.32i 669 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋)) |
9 | brcnvg 5303 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) | |
10 | 9 | ancoms 469 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
11 | 10 | bicomd 213 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥)) |
12 | 11 | notbid 308 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥)) |
13 | 12 | ralbidva 2985 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥)) |
14 | 13 | reubiia 3127 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥) |
15 | sup0 8372 | . . 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 1369 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, ◡𝑅) = 𝑋) |
17 | 1, 16 | syl5eq 2668 | 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃!wreu 2914 ∅c0 3915 class class class wbr 4653 Or wor 5034 ◡ccnv 5113 supcsup 8346 infcinf 8347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-po 5035 df-so 5036 df-cnv 5122 df-iota 5851 df-riota 6611 df-sup 8348 df-inf 8349 |
This theorem is referenced by: (None) |
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