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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 6822 | . 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 4676 | . . 3 ⊢ Ⅎ𝑥◡𝑅 |
6 | 2, 3, 5 | nfsup 6829 | . 2 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, ◡𝑅) |
7 | 1, 6 | nfcxfr 2250 | 1 ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 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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