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Theorem infmin 9532
Description: The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by AV, 3-Sep-2020.)
Hypotheses
Ref Expression
infmin.1 (𝜑𝑅 Or 𝐴)
infmin.2 (𝜑𝐶𝐴)
infmin.3 (𝜑𝐶𝐵)
infmin.4 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
Assertion
Ref Expression
infmin (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶   𝑦,𝑅   𝜑,𝑦

Proof of Theorem infmin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 infmin.1 . 2 (𝜑𝑅 Or 𝐴)
2 infmin.2 . 2 (𝜑𝐶𝐴)
3 infmin.4 . 2 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
4 infmin.3 . . 3 (𝜑𝐶𝐵)
5 simprr 773 . . 3 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6 breq1 5151 . . . 4 (𝑧 = 𝐶 → (𝑧𝑅𝑦𝐶𝑅𝑦))
76rspcev 3622 . . 3 ((𝐶𝐵𝐶𝑅𝑦) → ∃𝑧𝐵 𝑧𝑅𝑦)
84, 5, 7syl2an2r 685 . 2 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
91, 2, 3, 8eqinfd 9523 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068   class class class wbr 5148   Or wor 5596  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:  infpr  9541  lbinf  12219  uzinfi  12968  lcmgcdlem  16640  ramcl2lem  17043  oms0  34279  ballotlemirc  34513  inffz  35710  supinf  42262  oninfint  43225
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