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Theorem infnlbti 6642
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 6641 . . . . 5 (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))
43expdimp 255 . . . 4 ((𝜑𝐶𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ∃𝑧𝐵 𝑧𝑅𝐶))
5 rexalim 2369 . . . 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 1436  wral 2355  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: (None)
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