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Mathbox for Richard Penner |
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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 41761 | . . . . 5 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵) |
7 | 6 | sseld 3948 | . . . 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 3161 | . 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 846 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ⊆ wss 3915 ∅c0 4287 ifcif 4491 {csn 4591 class class class wbr 5110 Or wor 5549 1oc1o 8410 ≺ csdm 8889 Fincfn 8890 supcsup 9383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-om 7808 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 |
This theorem is referenced by: (None) |
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