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Theorem infempty 8397
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 8334 . 2 inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, 𝑅)
2 cnvso 5662 . . 3 (𝑅 Or 𝐴𝑅 Or 𝐴)
3 brcnvg 5292 . . . . . . . 8 ((𝑦𝐴𝑋𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
43ancoms 469 . . . . . . 7 ((𝑋𝐴𝑦𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
54bicomd 213 . . . . . 6 ((𝑋𝐴𝑦𝐴) → (𝑋𝑅𝑦𝑦𝑅𝑋))
65notbid 308 . . . . 5 ((𝑋𝐴𝑦𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦𝑅𝑋))
76ralbidva 2982 . . . 4 (𝑋𝐴 → (∀𝑦𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
87pm5.32i 668 . . 3 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
9 brcnvg 5292 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
109ancoms 469 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
1110bicomd 213 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑦𝑅𝑥))
1211notbid 308 . . . . 5 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
1312ralbidva 2982 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1413reubiia 3122 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
15 sup0 8357 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
162, 8, 14, 15syl3anb 1367 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, 𝑅) = 𝑋)
171, 16syl5eq 2666 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  ∃!wreu 2911  c0 3907   class class class wbr 4644   Or wor 5024  ccnv 5103  supcsup 8331  infcinf 8332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-po 5025  df-so 5026  df-cnv 5112  df-iota 5839  df-riota 6596  df-sup 8333  df-inf 8334
This theorem is referenced by: (None)
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