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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 42906 | . . . . 5 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵) |
7 | 6 | sseld 3972 | . . . 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 3153 | . 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 845 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ⊆ wss 3941 ∅c0 4319 ifcif 4525 {csn 4625 class class class wbr 5144 Or wor 5584 1oc1o 8473 ≺ csdm 8956 Fincfn 8957 supcsup 9458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-om 7866 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 |
This theorem is referenced by: (None) |
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