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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 43404 | . . . . 5 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵) |
7 | 6 | sseld 3993 | . . . 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 3160 | . 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 847 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ⊆ wss 3962 ∅c0 4338 ifcif 4530 {csn 4630 class class class wbr 5147 Or wor 5595 1oc1o 8497 ≺ csdm 8982 Fincfn 8983 supcsup 9477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-om 7887 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 |
This theorem is referenced by: (None) |
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