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Theorem infmin 8283
 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 . . . 4 (𝜑𝐶𝐵)
54adantr 480 . . 3 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → 𝐶𝐵)
6 simprr 792 . . 3 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
7 breq1 4586 . . . 4 (𝑧 = 𝐶 → (𝑧𝑅𝑦𝐶𝑅𝑦))
87rspcev 3282 . . 3 ((𝐶𝐵𝐶𝑅𝑦) → ∃𝑧𝐵 𝑧𝑅𝑦)
95, 6, 8syl2anc 691 . 2 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
101, 2, 3, 9eqinfd 8274 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583   Or wor 4958  infcinf 8230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-cnv 5046  df-iota 5768  df-riota 6511  df-sup 8231  df-inf 8232 This theorem is referenced by:  infpr  8292  uzinfi  11644  lcmgcdlem  15157  ramcl2lem  15551  oms0  29686  ballotlemirc  29920  inffz  30867
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