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Theorem infexd 9523
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 9483 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infexd.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 6308 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 218 . . 3 (𝜑𝑅 Or 𝐴)
54supexd 9493 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
61, 5eqeltrid 2845 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480   Or wor 5591  ccnv 5684  supcsup 9480  infcinf 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-cnv 5693  df-sup 9482  df-inf 9483
This theorem is referenced by:  infex  9533  omsfval  34296  wsucex  35827  prproropf1olem4  47493  prmdvdsfmtnof1  47574
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