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Theorem nfinf 6854
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 6822 . 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 4676 . . 3 𝑥𝑅
62, 3, 5nfsup 6829 . 2 𝑥sup(𝐴, 𝐵, 𝑅)
71, 6nfcxfr 2250 1 𝑥inf(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
Syntax hints:  wnfc 2240  ccnv 4496  supcsup 6819  infcinf 6820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-cnv 4505  df-sup 6821  df-inf 6822
This theorem is referenced by: (None)
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