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Mirrors > Home > MPE Home > Th. List > infmin | Structured version Visualization version 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 AV, 3-Sep-2020.) |
Ref | Expression |
---|---|
infmin.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
infmin.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
infmin.3 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
infmin.4 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) |
Ref | Expression |
---|---|
infmin | ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infmin.1 | . 2 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
2 | infmin.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
3 | infmin.4 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) | |
4 | infmin.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
5 | simprr 771 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦) | |
6 | breq1 5061 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
7 | 6 | rspcev 3622 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
8 | 4, 5, 7 | syl2an2r 683 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
9 | 1, 2, 3, 8 | eqinfd 8943 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5058 Or wor 5467 infcinf 8899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-po 5468 df-so 5469 df-cnv 5557 df-iota 6308 df-riota 7108 df-sup 8900 df-inf 8901 |
This theorem is referenced by: infpr 8961 lbinf 11588 uzinfi 12322 lcmgcdlem 15944 ramcl2lem 16339 oms0 31550 ballotlemirc 31784 inffz 32956 |
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