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Theorem infempty 8372
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 8309 . 2 inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, 𝑅)
2 cnvso 5643 . . 3 (𝑅 Or 𝐴𝑅 Or 𝐴)
3 brcnvg 5273 . . . . . . . 8 ((𝑦𝐴𝑋𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
43ancoms 469 . . . . . . 7 ((𝑋𝐴𝑦𝐴) → (𝑦𝑅𝑋𝑋𝑅𝑦))
54bicomd 213 . . . . . 6 ((𝑋𝐴𝑦𝐴) → (𝑋𝑅𝑦𝑦𝑅𝑋))
65notbid 308 . . . . 5 ((𝑋𝐴𝑦𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦𝑅𝑋))
76ralbidva 2981 . . . 4 (𝑋𝐴 → (∀𝑦𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
87pm5.32i 668 . . 3 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
9 brcnvg 5273 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
109ancoms 469 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑥𝑅𝑦))
1110bicomd 213 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑦𝑅𝑥))
1211notbid 308 . . . . 5 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
1312ralbidva 2981 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑥))
1413reubiia 3120 . . 3 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
15 sup0 8332 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
162, 8, 14, 15syl3anb 1366 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, 𝑅) = 𝑋)
171, 16syl5eq 2667 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 1480  wcel 1987  wral 2908  ∃!wreu 2910  c0 3897   class class class wbr 4623   Or wor 5004  ccnv 5083  supcsup 8306  infcinf 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-po 5005  df-so 5006  df-cnv 5092  df-iota 5820  df-riota 6576  df-sup 8308  df-inf 8309
This theorem is referenced by: (None)
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