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| Mirrors > Home > MPE Home > Th. List > infltoreq | Structured version Visualization version GIF version | ||
| Description: The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.) |
| Ref | Expression |
|---|---|
| infltoreq.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
| infltoreq.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| infltoreq.3 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
| infltoreq.4 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
| infltoreq.5 | ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅)) |
| Ref | Expression |
|---|---|
| infltoreq | ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | infltoreq.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
| 2 | cnvso 6308 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
| 3 | 1, 2 | sylib 218 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
| 4 | infltoreq.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 5 | infltoreq.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
| 6 | infltoreq.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
| 7 | infltoreq.5 | . . . 4 ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅)) | |
| 8 | df-inf 9483 | . . . 4 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
| 9 | 7, 8 | eqtrdi 2793 | . . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, ◡𝑅)) |
| 10 | 3, 4, 5, 6, 9 | supgtoreq 9510 | . 2 ⊢ (𝜑 → (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆)) |
| 11 | 6 | ne0d 4342 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
| 12 | fiinfcl 9541 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
| 13 | 1, 5, 11, 4, 12 | syl13anc 1374 | . . . . 5 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
| 14 | 7, 13 | eqeltrd 2841 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
| 15 | brcnvg 5890 | . . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝐶◡𝑅𝑆 ↔ 𝑆𝑅𝐶)) | |
| 16 | 15 | bicomd 223 | . . . 4 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
| 17 | 6, 14, 16 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
| 18 | 17 | orbi1d 917 | . 2 ⊢ (𝜑 → ((𝑆𝑅𝐶 ∨ 𝐶 = 𝑆) ↔ (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆))) |
| 19 | 10, 18 | mpbird 257 | 1 ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 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: (None) |
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