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Theorem infexd 9480
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 9440 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infexd.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 6281 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 217 . . 3 (𝜑𝑅 Or 𝐴)
54supexd 9450 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
61, 5eqeltrid 2831 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3468   Or wor 5580  ccnv 5668  supcsup 9437  infcinf 9438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-po 5581  df-so 5582  df-cnv 5677  df-sup 9439  df-inf 9440
This theorem is referenced by:  infex  9490  omsfval  33823  wsucex  35331  prproropf1olem4  46751  prmdvdsfmtnof1  46832
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