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Theorem nfinf 8676
 Description: Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
Hypotheses
Ref Expression
nfinf.1 𝑥𝐴
nfinf.2 𝑥𝐵
nfinf.3 𝑥𝑅
Assertion
Ref Expression
nfinf 𝑥inf(𝐴, 𝐵, 𝑅)

Proof of Theorem nfinf
StepHypRef Expression
1 df-inf 8637 . 2 inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
2 nfinf.1 . . 3 𝑥𝐴
3 nfinf.2 . . 3 𝑥𝐵
4 nfinf.3 . . . 4 𝑥𝑅
54nfcnv 5546 . . 3 𝑥𝑅
62, 3, 5nfsup 8645 . 2 𝑥sup(𝐴, 𝐵, 𝑅)
71, 6nfcxfr 2931 1 𝑥inf(𝐴, 𝐵, 𝑅)
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnfc 2918  ◡ccnv 5354  supcsup 8634  infcinf 8635 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-sup 8636  df-inf 8637 This theorem is referenced by:  iundisj  23752  iundisjf  29979  iundisjfi  30133  nfwsuc  32366  nfwlim  32370  allbutfiinf  40545  infrpgernmpt  40592  liminflelimsuplem  40907  stoweidlem62  41198  fourierdlem31  41274  iunhoiioolem  41808  smfinf  41943  prmdvdsfmtnof1lem1  42509
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