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Theorem inflb 8355
Description: An infimum is a lower bound. See also infcl 8354 and infglb 8356. (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 5643 . . . . . 6 (𝑅 Or 𝐴𝑅 Or 𝐴)
31, 2sylib 208 . . . . 5 (𝜑𝑅 Or 𝐴)
4 infcl.2 . . . . . 6 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
51, 4infcllem 8353 . . . . 5 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
63, 5supub 8325 . . . 4 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
76imp 445 . . 3 ((𝜑𝐶𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
8 df-inf 8309 . . . . . 6 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
98a1i 11 . . . . 5 ((𝜑𝐶𝐵) → inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅))
109breq2d 4635 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
113, 5supcl 8324 . . . . 5 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
12 brcnvg 5273 . . . . . 6 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝐶𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1312bicomd 213 . . . . 5 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1411, 13sylan 488 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1510, 14bitrd 268 . . 3 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
167, 15mtbird 315 . 2 ((𝜑𝐶𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))
1716ex 450 1 (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909   class class class wbr 4623   Or wor 5004  ccnv 5083  supcsup 8306  infcinf 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-po 5005  df-so 5006  df-cnv 5092  df-iota 5820  df-riota 6576  df-sup 8308  df-inf 8309
This theorem is referenced by:  infrelb  10968  infxrlb  12123  infssd  29372  infxrge0lb  29414  omssubadd  30185  ballotlemimin  30390  ballotlemfrcn0  30414  wsuclb  31531
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