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Theorem inflb 9512
Description: An infimum is a lower bound. See also infcl 9511 and infglb 9513. (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 6292 . . . . . 6 (𝑅 Or 𝐴𝑅 Or 𝐴)
31, 2sylib 217 . . . . 5 (𝜑𝑅 Or 𝐴)
4 infcl.2 . . . . . 6 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
51, 4infcllem 9510 . . . . 5 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
63, 5supub 9482 . . . 4 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
76imp 406 . . 3 ((𝜑𝐶𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
8 df-inf 9466 . . . . . 6 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
98a1i 11 . . . . 5 ((𝜑𝐶𝐵) → inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅))
109breq2d 5160 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
113, 5supcl 9481 . . . . 5 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
12 brcnvg 5882 . . . . . 6 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝐶𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1312bicomd 222 . . . . 5 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1411, 13sylan 579 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1510, 14bitrd 279 . . 3 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
167, 15mtbird 325 . 2 ((𝜑𝐶𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))
1716ex 412 1 (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3058  wrex 3067   class class class wbr 5148   Or wor 5589  ccnv 5677  supcsup 9463  infcinf 9464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-po 5590  df-so 5591  df-cnv 5686  df-iota 6500  df-riota 7376  df-sup 9465  df-inf 9466
This theorem is referenced by:  infrelb  12229  infxrlb  13345  infssd  32492  infxrge0lb  32534  omssubadd  33920  ballotlemimin  34125  ballotlemfrcn0  34149  wsuclb  35424
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