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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 4251 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
7 | fisup2g 8916 | . . . . . . . 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 1369 | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
9 | ssrexv 3982 | . . . . . . 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 8907 | . . . . 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 5048 | . . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶) |
14 | fisupcl 8917 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
15 | 3, 5, 6, 4, 14 | syl13anc 1369 | . . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
16 | 4, 15 | sseldd 3916 | . . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
17 | 1, 16 | eqeltrd 2890 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
18 | 4, 2 | sseldd 3916 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
19 | sotric 5465 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) | |
20 | 3, 17, 18, 19 | syl12anc 835 | . . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) |
21 | orcom 867 | . . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶)) | |
22 | eqcom 2805 | . . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆) | |
23 | 22 | orbi2i 910 | . . . . . 6 ⊢ ((𝐶𝑅𝑆 ∨ 𝑆 = 𝐶) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
24 | 21, 23 | bitri 278 | . . . . 5 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
25 | 24 | notbii 323 | . . . 4 ⊢ (¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
26 | 20, 25 | syl6rbb 291 | . . 3 ⊢ (𝜑 → (¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆) ↔ 𝑆𝑅𝐶)) |
27 | 13, 26 | mtbird 328 | . 2 ⊢ (𝜑 → ¬ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
28 | 27 | notnotrd 135 | 1 ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 Or wor 5437 Fincfn 8492 supcsup 8888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-om 7561 df-1o 8085 df-er 8272 df-en 8493 df-fin 8496 df-sup 8890 |
This theorem is referenced by: infltoreq 8950 supfirege 11614 |
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