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Theorem infminti 6931
 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 Jim Kingdon, 18-Dec-2021.)
Hypotheses
Ref Expression
infminti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
infminti.2 (𝜑𝐶𝐴)
infminti.3 (𝜑𝐶𝐵)
infminti.4 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
Assertion
Ref Expression
infminti (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Distinct variable groups:   𝑢,𝐴,𝑣,𝑦   𝑢,𝐵,𝑣,𝑦   𝑢,𝐶,𝑣,𝑦   𝑢,𝑅,𝑣,𝑦   𝜑,𝑢,𝑣,𝑦

Proof of Theorem infminti
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 infminti.ti . 2 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2 infminti.2 . 2 (𝜑𝐶𝐴)
3 infminti.4 . 2 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
4 infminti.3 . . 3 (𝜑𝐶𝐵)
5 simprr 522 . . 3 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6 breq1 3942 . . . 4 (𝑧 = 𝐶 → (𝑧𝑅𝑦𝐶𝑅𝑦))
76rspcev 2795 . . 3 ((𝐶𝐵𝐶𝑅𝑦) → ∃𝑧𝐵 𝑧𝑅𝑦)
84, 5, 7syl2an2r 585 . 2 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
91, 2, 3, 8eqinftid 6925 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2112  ∃wrex 2419   class class class wbr 3939  infcinf 6887 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-reu 2425  df-rmo 2426  df-rab 2427  df-v 2693  df-sbc 2916  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-br 3940  df-opab 4000  df-cnv 4559  df-iota 5100  df-riota 5742  df-sup 6888  df-inf 6889 This theorem is referenced by:  lbinf  8759  lcmgcdlem  11830  pilem3  12948  inffz  13483  taupi  13484
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