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Theorem nfinf 7045
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 7013 . 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 4824 . . 3 𝑥𝑅
62, 3, 5nfsup 7020 . 2 𝑥sup(𝐴, 𝐵, 𝑅)
71, 6nfcxfr 2329 1 𝑥inf(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
Syntax hints:  wnfc 2319  ccnv 4643  supcsup 7010  infcinf 7011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-cnv 4652  df-sup 7012  df-inf 7013
This theorem is referenced by: (None)
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