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Theorem infcl 9423
Description: An infimum belongs to its base class (closure law). See also inflb 9424 and infglb 9425. (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 9378 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infcl.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 6240 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 217 . . 3 (𝜑𝑅 Or 𝐴)
5 infcl.2 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
62, 5infcllem 9422 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
74, 6supcl 9393 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
81, 7eqeltrid 2842 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105   Or wor 5544  ccnv 5632  supcsup 9375  infcinf 9376
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-po 5545  df-so 5546  df-cnv 5641  df-iota 6448  df-riota 7312  df-sup 9377  df-inf 9378
This theorem is referenced by:  infrecl  12136  infxrcl  13251  infssd  31622  xrge0infssd  31661  infxrge0lb  31664  infxrge0gelb  31666  omsf  32887  wzel  34390  wsuccl  34393
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