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Theorem infexd 8386
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 8346 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 infexd.1 . . . 4 (𝜑𝑅 Or 𝐴)
3 cnvso 5672 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
42, 3sylib 208 . . 3 (𝜑𝑅 Or 𝐴)
54supexd 8356 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
61, 5syl5eqel 2704 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1989  Vcvv 3198   Or wor 5032  ccnv 5111  supcsup 8343  infcinf 8344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rmo 2919  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-po 5033  df-so 5034  df-cnv 5120  df-sup 8345  df-inf 8346
This theorem is referenced by:  infex  8396  omsfval  30341  wsucex  31759  prmdvdsfmtnof1  41270
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