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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 769 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦) | |
6 | breq1 4965 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
7 | 6 | rspcev 3559 | . . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
8 | 4, 5, 7 | syl2an2r 681 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
9 | 1, 2, 3, 8 | eqinfd 8795 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 class class class wbr 4962 Or wor 5361 infcinf 8751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-po 5362 df-so 5363 df-cnv 5451 df-iota 6189 df-riota 6977 df-sup 8752 df-inf 8753 |
This theorem is referenced by: infpr 8813 lbinf 11442 uzinfi 12177 lcmgcdlem 15779 ramcl2lem 16174 oms0 31172 ballotlemirc 31406 inffz 32569 |
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