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Theorem infcl 9529
Description: An infimum belongs to its base class (closure law). See also inflb 9530 and infglb 9531. (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 9484 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infcl.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 6307 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 218 . . 3 (𝜑𝑅 Or 𝐴)
5 infcl.2 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
62, 5infcllem 9528 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
74, 6supcl 9499 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
81, 7eqeltrid 2844 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2107  wral 3060  wrex 3069   class class class wbr 5142   Or wor 5590  ccnv 5683  supcsup 9481  infcinf 9482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-po 5591  df-so 5592  df-cnv 5692  df-iota 6513  df-riota 7389  df-sup 9483  df-inf 9484
This theorem is referenced by:  infrecl  12251  infxrcl  13376  infssd  32723  xrge0infssd  32766  infxrge0lb  32769  infxrge0gelb  32771  omsf  34299  wzel  35826  wsuccl  35829
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