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Theorem infempty 9547
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.)
Assertion
Ref Expression
infempty ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦

Proof of Theorem infempty
StepHypRef Expression
1 df-inf 9483 . 2 inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, 𝑅)
2 cnvso 6308 . . 3 (𝑅 Or 𝐴𝑅 Or 𝐴)
3 brcnvg 5890 . . . . . . . 8 ((𝑦𝐴𝑋𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
43ancoms 458 . . . . . . 7 ((𝑋𝐴𝑦𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
54bicomd 223 . . . . . 6 ((𝑋𝐴𝑦𝐴) → (𝑋𝑅𝑦𝑦𝑅𝑋))
65notbid 318 . . . . 5 ((𝑋𝐴𝑦𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦𝑅𝑋))
76ralbidva 3176 . . . 4 (𝑋𝐴 → (∀𝑦𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
87pm5.32i 574 . . 3 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
9 brcnvg 5890 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
109ancoms 458 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
1110bicomd 223 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑦𝑅𝑥))
1211notbid 318 . . . . 5 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
1312ralbidva 3176 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1413reubiia 3387 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
15 sup0 9506 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
162, 8, 14, 15syl3anb 1162 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, 𝑅) = 𝑋)
171, 16eqtrid 2789 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  ∃!wreu 3378  c0 4333   class class class wbr 5143   Or wor 5591  ccnv 5684  supcsup 9480  infcinf 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-cnv 5693  df-iota 6514  df-riota 7388  df-sup 9482  df-inf 9483
This theorem is referenced by: (None)
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