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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 4069 | . . . . . . . . 9 ⊢ (𝐶 ∈ 𝐵 → 𝐵 ≠ ∅) | |
6 | 1, 5 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅) |
7 | fisup2g 8530 | . . . . . . . 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 1478 | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
9 | ssrexv 3816 | . . . . . . 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 65 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
11 | 2, 10 | supub 8521 | . . . . 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 4796 | . . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) |
15 | 12, 14 | mtbird 314 | . . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶) |
16 | fisupcl 8531 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
17 | 2, 4, 6, 3, 16 | syl13anc 1478 | . . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
18 | 3, 17 | sseldd 3753 | . . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
19 | 13, 18 | eqeltrd 2850 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
20 | 3, 1 | sseldd 3753 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
21 | sotric 5196 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) | |
22 | 2, 19, 20, 21 | syl12anc 1474 | . . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆))) |
23 | orcom 857 | . . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶)) | |
24 | eqcom 2778 | . . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆) | |
25 | 24 | orbi2i 896 | . . . . . 6 ⊢ ((𝐶𝑅𝑆 ∨ 𝑆 = 𝐶) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
26 | 23, 25 | bitri 264 | . . . . 5 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
27 | 26 | notbii 309 | . . . 4 ⊢ (¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
28 | 22, 27 | syl6rbb 277 | . . 3 ⊢ (𝜑 → (¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆) ↔ 𝑆𝑅𝐶)) |
29 | 15, 28 | mtbird 314 | . 2 ⊢ (𝜑 → ¬ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
30 | 29 | notnotrd 130 | 1 ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∨ wo 834 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∀wral 3061 ∃wrex 3062 ⊆ wss 3723 ∅c0 4063 class class class wbr 4786 Or wor 5169 Fincfn 8109 supcsup 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-om 7213 df-1o 7713 df-er 7896 df-en 8110 df-fin 8113 df-sup 8504 |
This theorem is referenced by: infltoreq 8564 supfirege 11211 |
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