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Theorem infminti 6643
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 499 . . 3 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6 breq1 3817 . . . 4 (𝑧 = 𝐶 → (𝑧𝑅𝑦𝐶𝑅𝑦))
76rspcev 2714 . . 3 ((𝐶𝐵𝐶𝑅𝑦) → ∃𝑧𝐵 𝑧𝑅𝑦)
84, 5, 7syl2an2r 560 . 2 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
91, 2, 3, 8eqinftid 6637 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  wrex 2356   class class class wbr 3814  infcinf 6599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-cnv 4412  df-iota 4937  df-riota 5550  df-sup 6600  df-inf 6601
This theorem is referenced by:  lbinf  8321  lcmgcdlem  10853
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