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Mirrors > Home > ILE Home > Th. List > infsnti | GIF version |
Description: The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.) |
Ref | Expression |
---|---|
infsnti.ti | ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) |
infsnti.b | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
infsnti | ⊢ (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6872 | . 2 ⊢ inf({𝐵}, 𝐴, 𝑅) = sup({𝐵}, 𝐴, ◡𝑅) | |
2 | infsnti.ti | . . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) | |
3 | 2 | cnvti 6906 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢))) |
4 | infsnti.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
5 | 3, 4 | supsnti 6892 | . 2 ⊢ (𝜑 → sup({𝐵}, 𝐴, ◡𝑅) = 𝐵) |
6 | 1, 5 | syl5eq 2184 | 1 ⊢ (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {csn 3527 class class class wbr 3929 ◡ccnv 4538 supcsup 6869 infcinf 6870 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-cnv 4547 df-iota 5088 df-riota 5730 df-sup 6871 df-inf 6872 |
This theorem is referenced by: (None) |
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