![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fimaxre4 | Structured version Visualization version GIF version |
Description: A nonempty finite set of real numbers is bounded (image set version). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fimaxre4.1 | ⊢ Ⅎ𝑥𝜑 |
fimaxre4.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fimaxre4.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
fimaxre4 | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fimaxre4.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fimaxre4.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | fimaxre4.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 3 | ex 403 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ ℝ)) |
5 | 2, 4 | ralrimi 3139 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ) |
6 | fimaxre3 11328 | . 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) | |
7 | 1, 5, 6 | syl2anc 579 | 1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 Ⅎwnf 1827 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 class class class wbr 4888 Fincfn 8243 ℝcr 10273 ≤ cle 10414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-addrcl 10335 ax-rnegex 10345 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 |
This theorem is referenced by: limsupubuzlem 40866 |
Copyright terms: Public domain | W3C validator |