Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  infmin Structured version   Visualization version   GIF version

Theorem infmin 8945
 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 772 . . 3 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6 breq1 5034 . . . 4 (𝑧 = 𝐶 → (𝑧𝑅𝑦𝐶𝑅𝑦))
76rspcev 3571 . . 3 ((𝐶𝐵𝐶𝑅𝑦) → ∃𝑧𝐵 𝑧𝑅𝑦)
84, 5, 7syl2an2r 684 . 2 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
91, 2, 3, 8eqinfd 8936 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   class class class wbr 5031   Or wor 5438  infcinf 8892 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-po 5439  df-so 5440  df-cnv 5528  df-iota 6284  df-riota 7094  df-sup 8893  df-inf 8894 This theorem is referenced by:  infpr  8954  lbinf  11584  uzinfi  12319  lcmgcdlem  15943  ramcl2lem  16338  oms0  31680  ballotlemirc  31914  inffz  33089
 Copyright terms: Public domain W3C validator