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Theorem infcl 8940
Description: An infimum belongs to its base class (closure law). See also inflb 8941 and infglb 8942. (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
infcl (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem infcl
StepHypRef Expression
1 df-inf 8895 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infcl.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 6117 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 221 . . 3 (𝜑𝑅 Or 𝐴)
5 infcl.2 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
62, 5infcllem 8939 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
74, 6supcl 8910 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
81, 7eqeltrid 2918 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wcel 2114  wral 3130  wrex 3131   class class class wbr 5042   Or wor 5450  ccnv 5531  supcsup 8892  infcinf 8893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-po 5451  df-so 5452  df-cnv 5540  df-iota 6293  df-riota 7098  df-sup 8894  df-inf 8895
This theorem is referenced by:  infrecl  11610  infxrcl  12714  infssd  30456  xrge0infssd  30495  infxrge0lb  30498  infxrge0gelb  30500  omsf  31628  wzel  33185  wsuccl  33188
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