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Theorem nfinf 9483
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 9444 . 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 5878 . . 3 𝑥𝑅
62, 3, 5nfsup 9452 . 2 𝑥sup(𝐴, 𝐵, 𝑅)
71, 6nfcxfr 2900 1 𝑥inf(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2882  ccnv 5675  supcsup 9441  infcinf 9442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-sup 9443  df-inf 9444
This theorem is referenced by:  iundisj  25397  iundisjf  32253  iundisjfi  32440  nfwsuc  35260  nfwlim  35264  allbutfiinf  44589  infrpgernmpt  44634  liminflelimsuplem  44950  stoweidlem62  45237  fourierdlem31  45313  iunhoiioolem  45850  smfinf  45993  prmdvdsfmtnof1lem1  46711
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