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Theorem nfinf 8924
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 8885 . 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 5725 . . 3 𝑥𝑅
62, 3, 5nfsup 8893 . 2 𝑥sup(𝐴, 𝐵, 𝑅)
71, 6nfcxfr 2971 1 𝑥inf(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2957  ccnv 5530  supcsup 8882  infcinf 8883
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-xp 5537  df-cnv 5539  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-sup 8884  df-inf 8885
This theorem is referenced by:  iundisj  24131  iundisjf  30326  iundisjfi  30506  nfwsuc  33113  nfwlim  33117  allbutfiinf  41848  infrpgernmpt  41895  liminflelimsuplem  42208  stoweidlem62  42495  fourierdlem31  42571  iunhoiioolem  43105  smfinf  43240  prmdvdsfmtnof1lem1  43892
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