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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 42758 | . . . . 5 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵) |
7 | 6 | sseld 3977 | . . . 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 3158 | . 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 1534 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 ∅c0 4318 ifcif 4524 {csn 4624 class class class wbr 5142 Or wor 5583 1oc1o 8471 ≺ csdm 8952 Fincfn 8953 supcsup 9449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-om 7863 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 |
This theorem is referenced by: (None) |
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