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Theorem infexd 9521
Description: An infimum is a set. (Contributed by AV, 2-Sep-2020.)
Hypothesis
Ref Expression
infexd.1 (𝜑𝑅 Or 𝐴)
Assertion
Ref Expression
infexd (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)

Proof of Theorem infexd
StepHypRef Expression
1 df-inf 9481 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infexd.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 6310 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 218 . . 3 (𝜑𝑅 Or 𝐴)
54supexd 9491 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
61, 5eqeltrid 2843 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478   Or wor 5596  ccnv 5688  supcsup 9478  infcinf 9479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-po 5597  df-so 5598  df-cnv 5697  df-sup 9480  df-inf 9481
This theorem is referenced by:  infex  9531  omsfval  34276  wsucex  35808  prproropf1olem4  47431  prmdvdsfmtnof1  47512
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