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Theorem infmin 8804
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 769 . . 3 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6 breq1 4965 . . . 4 (𝑧 = 𝐶 → (𝑧𝑅𝑦𝐶𝑅𝑦))
76rspcev 3559 . . 3 ((𝐶𝐵𝐶𝑅𝑦) → ∃𝑧𝐵 𝑧𝑅𝑦)
84, 5, 7syl2an2r 681 . 2 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
91, 2, 3, 8eqinfd 8795 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wcel 2081  wrex 3106   class class class wbr 4962   Or wor 5361  infcinf 8751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-po 5362  df-so 5363  df-cnv 5451  df-iota 6189  df-riota 6977  df-sup 8752  df-inf 8753
This theorem is referenced by:  infpr  8813  lbinf  11442  uzinfi  12177  lcmgcdlem  15779  ramcl2lem  16174  oms0  31172  ballotlemirc  31406  inffz  32569
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