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Theorem infempty 8959
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 8895 . 2 inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, 𝑅)
2 cnvso 6111 . . 3 (𝑅 Or 𝐴𝑅 Or 𝐴)
3 brcnvg 5718 . . . . . . . 8 ((𝑦𝐴𝑋𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
43ancoms 462 . . . . . . 7 ((𝑋𝐴𝑦𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
54bicomd 226 . . . . . 6 ((𝑋𝐴𝑦𝐴) → (𝑋𝑅𝑦𝑦𝑅𝑋))
65notbid 321 . . . . 5 ((𝑋𝐴𝑦𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦𝑅𝑋))
76ralbidva 3164 . . . 4 (𝑋𝐴 → (∀𝑦𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
87pm5.32i 578 . . 3 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
9 brcnvg 5718 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
109ancoms 462 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
1110bicomd 226 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑦𝑅𝑥))
1211notbid 321 . . . . 5 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
1312ralbidva 3164 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1413reubiia 3346 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
15 sup0 8918 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
162, 8, 14, 15syl3anb 1158 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, 𝑅) = 𝑋)
171, 16syl5eq 2848 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  ∃!wreu 3111  c0 4246   class class class wbr 5033   Or wor 5441  ccnv 5522  supcsup 8892  infcinf 8893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-po 5442  df-so 5443  df-cnv 5531  df-iota 6287  df-riota 7097  df-sup 8894  df-inf 8895
This theorem is referenced by: (None)
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