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Theorem nfinf 6872
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 6840 . 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 4688 . . 3  |-  F/_ x `' R
62, 3, 5nfsup 6847 . 2  |-  F/_ x sup ( A ,  B ,  `' R )
71, 6nfcxfr 2255 1  |-  F/_ xinf ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2245   `'ccnv 4508   supcsup 6837  infcinf 6838
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-cnv 4517  df-sup 6839  df-inf 6840
This theorem is referenced by: (None)
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