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Mirrors > Home > ILE Home > Th. List > nfinf | GIF version |
Description: Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
nfinf.1 | ⊢ Ⅎ𝑥𝐴 |
nfinf.2 | ⊢ Ⅎ𝑥𝐵 |
nfinf.3 | ⊢ Ⅎ𝑥𝑅 |
Ref | Expression |
---|---|
nfinf | ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6840 | . 2 ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) | |
2 | nfinf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfinf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | nfinf.3 | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
5 | 4 | nfcnv 4688 | . . 3 ⊢ Ⅎ𝑥◡𝑅 |
6 | 2, 3, 5 | nfsup 6847 | . 2 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, ◡𝑅) |
7 | 1, 6 | nfcxfr 2255 | 1 ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 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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