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Theorem infexd 8946
 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 8906 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infexd.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 6128 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 221 . . 3 (𝜑𝑅 Or 𝐴)
54supexd 8916 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
61, 5eqeltrid 2920 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  Vcvv 3480   Or wor 5461  ◡ccnv 5542  supcsup 8903  infcinf 8904 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rmo 3141  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-po 5462  df-so 5463  df-cnv 5551  df-sup 8905  df-inf 8906 This theorem is referenced by:  infex  8956  omsfval  31637  wsucex  33198  prproropf1olem4  43976  prmdvdsfmtnof1  44057
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