ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfinf Unicode version

Theorem nfinf 6976
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 6944 . 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 4780 . . 3  |-  F/_ x `' R
62, 3, 5nfsup 6951 . 2  |-  F/_ x sup ( A ,  B ,  `' R )
71, 6nfcxfr 2303 1  |-  F/_ xinf ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2293   `'ccnv 4600   supcsup 6941  infcinf 6942
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-un 3118  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-cnv 4609  df-sup 6943  df-inf 6944
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator