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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 6941 | . 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 6975 | . . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢))) |
4 | infsnti.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
5 | 3, 4 | supsnti 6961 | . 2 ⊢ (𝜑 → sup({𝐵}, 𝐴, ◡𝑅) = 𝐵) |
6 | 1, 5 | syl5eq 2209 | 1 ⊢ (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 {csn 3570 class class class wbr 3976 ◡ccnv 4597 supcsup 6938 infcinf 6939 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-cnv 4606 df-iota 5147 df-riota 5792 df-sup 6940 df-inf 6941 |
This theorem is referenced by: (None) |
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