| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fiinfcl | Structured version Visualization version GIF version | ||
| Description: A nonempty finite set contains its infimum. (Contributed by AV, 3-Sep-2020.) |
| Ref | Expression |
|---|---|
| fiinfcl | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-inf 9483 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
| 2 | cnvso 6308 | . . 3 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
| 3 | fisupcl 9509 | . . 3 ⊢ ((◡𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐵) | |
| 4 | 2, 3 | sylanb 581 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐵) |
| 5 | 1, 4 | eqeltrid 2845 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 Or wor 5591 ◡ccnv 5684 Fincfn 8985 supcsup 9480 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 ax-un 7755 |
| 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-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-om 7888 df-en 8986 df-fin 8989 df-sup 9482 df-inf 9483 |
| This theorem is referenced by: infltoreq 9542 aalioulem2 26375 ballotlemiex 34504 ptrecube 37627 heicant 37662 aks4d1p4 42080 aks4d1p7 42084 sticksstones1 42147 cnrefiisplem 45844 fourierdlem42 46164 ioorrnopnlem 46319 hoidmvlelem2 46611 iunhoiioolem 46690 vonioolem1 46695 prmdvdsfmtnof1lem1 47571 prmdvdsfmtnof 47573 |
| Copyright terms: Public domain | W3C validator |