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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 772 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦) | |
6 | breq1 5145 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
7 | 6 | rspcev 3607 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
8 | 4, 5, 7 | syl2an2r 684 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
9 | 1, 2, 3, 8 | eqinfd 9500 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 class class class wbr 5142 Or wor 5583 infcinf 9456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-po 5584 df-so 5585 df-cnv 5680 df-iota 6494 df-riota 7370 df-sup 9457 df-inf 9458 |
This theorem is referenced by: infpr 9518 lbinf 12189 uzinfi 12934 lcmgcdlem 16568 ramcl2lem 16969 oms0 33853 ballotlemirc 34087 inffz 35260 oninfint 42587 |
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