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Theorem inflb 8748
Description: An infimum is a lower bound. See also infcl 8747 and infglb 8749. (Contributed by AV, 3-Sep-2020.)
Hypotheses
Ref Expression
infcl.1 (𝜑𝑅 Or 𝐴)
infcl.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
Assertion
Ref Expression
inflb (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem inflb
StepHypRef Expression
1 infcl.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
2 cnvso 5977 . . . . . 6 (𝑅 Or 𝐴𝑅 Or 𝐴)
31, 2sylib 210 . . . . 5 (𝜑𝑅 Or 𝐴)
4 infcl.2 . . . . . 6 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
51, 4infcllem 8746 . . . . 5 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
63, 5supub 8718 . . . 4 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
76imp 398 . . 3 ((𝜑𝐶𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
8 df-inf 8702 . . . . . 6 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
98a1i 11 . . . . 5 ((𝜑𝐶𝐵) → inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅))
109breq2d 4941 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
113, 5supcl 8717 . . . . 5 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
12 brcnvg 5600 . . . . . 6 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝐶𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1312bicomd 215 . . . . 5 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1411, 13sylan 572 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1510, 14bitrd 271 . . 3 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
167, 15mtbird 317 . 2 ((𝜑𝐶𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))
1716ex 405 1 (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wral 3089  wrex 3090   class class class wbr 4929   Or wor 5325  ccnv 5406  supcsup 8699  infcinf 8700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-po 5326  df-so 5327  df-cnv 5415  df-iota 6152  df-riota 6937  df-sup 8701  df-inf 8702
This theorem is referenced by:  infrelb  11427  infxrlb  12543  infssd  30198  infxrge0lb  30240  omssubadd  31200  ballotlemimin  31406  ballotlemfrcn0  31430  wsuclb  32633
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