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Mirrors > Home > ILE Home > Th. List > infminti | GIF version |
Description: The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infminti.ti | ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) |
infminti.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
infminti.3 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
infminti.4 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) |
Ref | Expression |
---|---|
infminti | ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infminti.ti | . 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) | |
2 | infminti.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
3 | infminti.4 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) | |
4 | infminti.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
5 | simprr 522 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦) | |
6 | breq1 3979 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
7 | 6 | rspcev 2825 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
8 | 4, 5, 7 | syl2an2r 585 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
9 | 1, 2, 3, 8 | eqinftid 6977 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 class class class wbr 3976 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: lbinf 8834 lcmgcdlem 11988 pilem3 13251 inffz 13789 taupi 13790 |
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