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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 7013 | . 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 4824 | . . 3 ⊢ Ⅎ𝑥◡𝑅 |
6 | 2, 3, 5 | nfsup 7020 | . 2 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, ◡𝑅) |
7 | 1, 6 | nfcxfr 2329 | 1 ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2319 ◡ccnv 4643 supcsup 7010 infcinf 7011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-cnv 4652 df-sup 7012 df-inf 7013 |
This theorem is referenced by: (None) |
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