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| Mirrors > Home > MPE Home > Th. List > Mathboxes > safesnsupfiub | Structured version Visualization version GIF version | ||
| Description: If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.) |
| Ref | Expression |
|---|---|
| safesnsupfiub.small | ⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o)) |
| safesnsupfiub.finite | ⊢ (𝜑 → 𝐵 ∈ Fin) |
| safesnsupfiub.subset | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| safesnsupfiub.ordered | ⊢ (𝜑 → 𝑅 Or 𝐴) |
| safesnsupfiub.ub | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥𝑅𝑦) |
| Ref | Expression |
|---|---|
| safesnsupfiub | ⊢ (𝜑 → ∀𝑥 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦 ∈ 𝐶 𝑥𝑅𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | safesnsupfiub.ub | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥𝑅𝑦) | |
| 2 | safesnsupfiub.small | . . . . . 6 ⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o)) | |
| 3 | safesnsupfiub.finite | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
| 4 | safesnsupfiub.subset | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 5 | safesnsupfiub.ordered | . . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
| 6 | 2, 3, 4, 5 | safesnsupfiss 43428 | . . . . 5 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵) |
| 7 | 6 | sseld 3982 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) → 𝑥 ∈ 𝐵)) |
| 8 | 7 | imim1d 82 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 → ∀𝑦 ∈ 𝐶 𝑥𝑅𝑦) → (𝑥 ∈ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) → ∀𝑦 ∈ 𝐶 𝑥𝑅𝑦))) |
| 9 | 8 | ralimdv2 3163 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑥𝑅𝑦 → ∀𝑥 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦 ∈ 𝐶 𝑥𝑅𝑦)) |
| 10 | 1, 9 | mpd 15 | 1 ⊢ (𝜑 → ∀𝑥 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦 ∈ 𝐶 𝑥𝑅𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 class class class wbr 5143 Or wor 5591 1oc1o 8499 ≺ csdm 8984 Fincfn 8985 supcsup 9480 |
| 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-pow 5365 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-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 |
| This theorem is referenced by: (None) |
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