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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 773 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦) | |
| 6 | breq1 5146 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
| 7 | 6 | rspcev 3622 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
| 8 | 4, 5, 7 | syl2an2r 685 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
| 9 | 1, 2, 3, 8 | eqinfd 9525 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 Or wor 5591 infcinf 9481 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-cnv 5693 df-iota 6514 df-riota 7388 df-sup 9482 df-inf 9483 |
| This theorem is referenced by: infpr 9543 lbinf 12221 uzinfi 12970 lcmgcdlem 16643 ramcl2lem 17047 oms0 34299 ballotlemirc 34534 inffz 35730 supinf 42283 oninfint 43248 |
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