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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 4298 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
7 | fisup2g 8920 | . . . . . . . 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 1364 | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
9 | ssrexv 4031 | . . . . . . 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 8911 | . . . . 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 5075 | . . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶) |
14 | fisupcl 8921 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
15 | 3, 5, 6, 4, 14 | syl13anc 1364 | . . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
16 | 4, 15 | sseldd 3965 | . . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
17 | 1, 16 | eqeltrd 2910 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
18 | 4, 2 | sseldd 3965 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
19 | sotric 5494 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) | |
20 | 3, 17, 18, 19 | syl12anc 832 | . . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) |
21 | orcom 864 | . . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶)) | |
22 | eqcom 2825 | . . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆) | |
23 | 22 | orbi2i 906 | . . . . . 6 ⊢ ((𝐶𝑅𝑆 ∨ 𝑆 = 𝐶) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
24 | 21, 23 | bitri 276 | . . . . 5 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
25 | 24 | notbii 321 | . . . 4 ⊢ (¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
26 | 20, 25 | syl6rbb 289 | . . 3 ⊢ (𝜑 → (¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆) ↔ 𝑆𝑅𝐶)) |
27 | 13, 26 | mtbird 326 | . 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 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 ∅c0 4288 class class class wbr 5057 Or wor 5466 Fincfn 8497 supcsup 8892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-om 7570 df-1o 8091 df-er 8278 df-en 8498 df-fin 8501 df-sup 8894 |
This theorem is referenced by: infltoreq 8954 supfirege 11615 |
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