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Theorem nfinf 6904
Description: Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
Hypotheses
Ref Expression
nfinf.1  |-  F/_ x A
nfinf.2  |-  F/_ x B
nfinf.3  |-  F/_ x R
Assertion
Ref Expression
nfinf  |-  F/_ xinf ( A ,  B ,  R )

Proof of Theorem nfinf
StepHypRef Expression
1 df-inf 6872 . 2  |- inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )
2 nfinf.1 . . 3  |-  F/_ x A
3 nfinf.2 . . 3  |-  F/_ x B
4 nfinf.3 . . . 4  |-  F/_ x R
54nfcnv 4718 . . 3  |-  F/_ x `' R
62, 3, 5nfsup 6879 . 2  |-  F/_ x sup ( A ,  B ,  `' R )
71, 6nfcxfr 2278 1  |-  F/_ xinf ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2268   `'ccnv 4538   supcsup 6869  infcinf 6870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-cnv 4547  df-sup 6871  df-inf 6872
This theorem is referenced by: (None)
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