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Theorem infempty 9545
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 9481 . 2 inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, 𝑅)
2 cnvso 6310 . . 3 (𝑅 Or 𝐴𝑅 Or 𝐴)
3 brcnvg 5893 . . . . . . . 8 ((𝑦𝐴𝑋𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
43ancoms 458 . . . . . . 7 ((𝑋𝐴𝑦𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
54bicomd 223 . . . . . 6 ((𝑋𝐴𝑦𝐴) → (𝑋𝑅𝑦𝑦𝑅𝑋))
65notbid 318 . . . . 5 ((𝑋𝐴𝑦𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦𝑅𝑋))
76ralbidva 3174 . . . 4 (𝑋𝐴 → (∀𝑦𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
87pm5.32i 574 . . 3 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
9 brcnvg 5893 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
109ancoms 458 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
1110bicomd 223 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑦𝑅𝑥))
1211notbid 318 . . . . 5 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
1312ralbidva 3174 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1413reubiia 3385 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
15 sup0 9504 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
162, 8, 14, 15syl3anb 1160 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, 𝑅) = 𝑋)
171, 16eqtrid 2787 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 1086   = wceq 1537  wcel 2106  wral 3059  ∃!wreu 3376  c0 4339   class class class wbr 5148   Or wor 5596  ccnv 5688  supcsup 9478  infcinf 9479
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-po 5597  df-so 5598  df-cnv 5697  df-iota 6516  df-riota 7388  df-sup 9480  df-inf 9481
This theorem is referenced by: (None)
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