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Theorem infnlbti 6534
Description: A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
Hypotheses
Ref Expression
infclti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
infclti.ex (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
Assertion
Ref Expression
infnlbti (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑥,𝑦,𝑧   𝑢,𝐵,𝑣,𝑥,𝑦,𝑧   𝑢,𝑅,𝑣,𝑥,𝑦,𝑧   𝜑,𝑢,𝑣,𝑥,𝑦,𝑧   𝑧,𝐶
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem infnlbti
StepHypRef Expression
1 infclti.ti . . . . . 6 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2 infclti.ex . . . . . 6 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
31, 2infglbti 6533 . . . . 5 (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))
43expdimp 255 . . . 4 ((𝜑𝐶𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ∃𝑧𝐵 𝑧𝑅𝐶))
5 rexalim 2366 . . . 4 (∃𝑧𝐵 𝑧𝑅𝐶 → ¬ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶)
64, 5syl6 33 . . 3 ((𝜑𝐶𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ¬ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶))
76con2d 587 . 2 ((𝜑𝐶𝐴) → (∀𝑧𝐵 ¬ 𝑧𝑅𝐶 → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
87expimpd 355 1 (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wcel 1434  wral 2353  wrex 2354   class class class wbr 3806  infcinf 6491
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-cnv 4400  df-iota 4918  df-riota 5520  df-sup 6492  df-inf 6493
This theorem is referenced by: (None)
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