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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 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶 ∈ 𝐵) |
6 | simprr 792 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦) | |
7 | breq1 4586 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
8 | 7 | rspcev 3282 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
9 | 5, 6, 8 | syl2anc 691 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
10 | 1, 2, 3, 9 | eqinfd 8274 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 Or wor 4958 infcinf 8230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-po 4959 df-so 4960 df-cnv 5046 df-iota 5768 df-riota 6511 df-sup 8231 df-inf 8232 |
This theorem is referenced by: infpr 8292 uzinfi 11644 lcmgcdlem 15157 ramcl2lem 15551 oms0 29686 ballotlemirc 29920 inffz 30867 |
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