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| Mirrors > Home > MPE Home > Th. List > supgtoreq | Structured version Visualization version GIF version | ||
| Description: The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.) |
| Ref | Expression |
|---|---|
| supgtoreq.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
| supgtoreq.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| supgtoreq.3 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
| supgtoreq.4 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
| supgtoreq.5 | ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅)) |
| Ref | Expression |
|---|---|
| supgtoreq | ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | supgtoreq.5 | . . . 4 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅)) | |
| 2 | supgtoreq.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
| 3 | supgtoreq.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
| 4 | supgtoreq.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 5 | supgtoreq.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
| 6 | 2 | ne0d 4342 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
| 7 | fisup2g 9508 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | |
| 8 | 3, 5, 6, 4, 7 | syl13anc 1374 | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
| 9 | ssrexv 4053 | . . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))) | |
| 10 | 4, 8, 9 | sylc 65 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
| 11 | 3, 10 | supub 9499 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) |
| 12 | 2, 11 | mpd 15 | . . . 4 ⊢ (𝜑 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶) |
| 13 | 1, 12 | eqnbrtrd 5161 | . . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶) |
| 14 | fisupcl 9509 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
| 15 | 3, 5, 6, 4, 14 | syl13anc 1374 | . . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
| 16 | 4, 15 | sseldd 3984 | . . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
| 17 | 1, 16 | eqeltrd 2841 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
| 18 | 4, 2 | sseldd 3984 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
| 19 | sotric 5622 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) | |
| 20 | 3, 17, 18, 19 | syl12anc 837 | . . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) |
| 21 | orcom 871 | . . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶)) | |
| 22 | eqcom 2744 | . . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆) | |
| 23 | 22 | orbi2i 913 | . . . . . 6 ⊢ ((𝐶𝑅𝑆 ∨ 𝑆 = 𝐶) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
| 24 | 21, 23 | bitri 275 | . . . . 5 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
| 25 | 24 | notbii 320 | . . . 4 ⊢ (¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
| 26 | 20, 25 | bitr2di 288 | . . 3 ⊢ (𝜑 → (¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆) ↔ 𝑆𝑅𝐶)) |
| 27 | 13, 26 | mtbird 325 | . 2 ⊢ (𝜑 → ¬ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
| 28 | 27 | notnotrd 133 | 1 ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 Or wor 5591 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-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-en 8986 df-fin 8989 df-sup 9482 |
| This theorem is referenced by: infltoreq 9542 supfirege 12255 safesnsupfilb 43431 |
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