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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.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
2 | supgtoreq.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | supgtoreq.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
4 | supgtoreq.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
5 | ne0i 3880 | . . . . . . . . 9 ⊢ (𝐶 ∈ 𝐵 → 𝐵 ≠ ∅) | |
6 | 1, 5 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
7 | fisup2g 8257 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | |
8 | 2, 4, 6, 3, 7 | syl13anc 1320 | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
9 | ssrexv 3630 | . . . . . . 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 | 3, 8, 9 | sylc 63 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
11 | 2, 10 | supub 8248 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) |
12 | 1, 11 | mpd 15 | . . . 4 ⊢ (𝜑 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶) |
13 | supgtoreq.5 | . . . . 5 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅)) | |
14 | 13 | breq1d 4593 | . . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) |
15 | 12, 14 | mtbird 314 | . . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶) |
16 | fisupcl 8258 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
17 | 2, 4, 6, 3, 16 | syl13anc 1320 | . . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
18 | 3, 17 | sseldd 3569 | . . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
19 | 13, 18 | eqeltrd 2688 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
20 | 3, 1 | sseldd 3569 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
21 | sotric 4985 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) | |
22 | 2, 19, 20, 21 | syl12anc 1316 | . . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) |
23 | orcom 401 | . . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶)) | |
24 | eqcom 2617 | . . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆) | |
25 | 24 | orbi2i 540 | . . . . . 6 ⊢ ((𝐶𝑅𝑆 ∨ 𝑆 = 𝐶) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
26 | 23, 25 | bitri 263 | . . . . 5 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
27 | 26 | notbii 309 | . . . 4 ⊢ (¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
28 | 22, 27 | syl6rbb 276 | . . 3 ⊢ (𝜑 → (¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆) ↔ 𝑆𝑅𝐶)) |
29 | 15, 28 | mtbird 314 | . 2 ⊢ (𝜑 → ¬ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
30 | 29 | notnotrd 127 | 1 ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 Or wor 4958 Fincfn 7841 supcsup 8229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-fin 7845 df-sup 8231 |
This theorem is referenced by: infltoreq 8291 supfirege 10886 |
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