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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressioosup | Structured version Visualization version GIF version |
Description: If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ressioosup.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ressioosup.s | ⊢ 𝑆 = sup(𝐴, ℝ*, < ) |
ressioosup.n | ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) |
ressioosup.i | ⊢ 𝐼 = (-∞(,)𝑆) |
Ref | Expression |
---|---|
ressioosup | ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 11278 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ ∈ ℝ*) |
3 | ressioosup.s | . . . . . 6 ⊢ 𝑆 = sup(𝐴, ℝ*, < ) | |
4 | ressioosup.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
5 | ressxr 11265 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
6 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
7 | 4, 6 | sstrd 3992 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
9 | 8 | supxrcld 44258 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | 3, 9 | eqeltrid 2836 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℝ*) |
11 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
12 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sseldd 3983 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
14 | 13 | mnfltd 13111 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ < 𝑥) |
15 | 7 | sselda 3982 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
16 | supxrub 13310 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) | |
17 | 8, 12, 16 | syl2anc 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) |
18 | 3 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = sup(𝐴, ℝ*, < )) |
19 | 18 | eqcomd 2737 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) = 𝑆) |
20 | 17, 19 | breqtrd 5174 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ 𝑆) |
21 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆) | |
22 | 21 | eqcomd 2737 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑆 → 𝑆 = 𝑥) |
23 | 22 | adantl 481 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 = 𝑥) |
24 | simpl 482 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑥 ∈ 𝐴) | |
25 | 23, 24 | eqeltrd 2832 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
26 | 25 | adantll 711 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
27 | ressioosup.n | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) | |
28 | 27 | ad2antrr 723 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → ¬ 𝑆 ∈ 𝐴) |
29 | 26, 28 | pm2.65da 814 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝑆) |
30 | 29 | neqned 2946 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝑆) |
31 | 15, 10, 20, 30 | xrleneltd 44492 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 < 𝑆) |
32 | 2, 10, 13, 14, 31 | eliood 44670 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (-∞(,)𝑆)) |
33 | ressioosup.i | . . . 4 ⊢ 𝐼 = (-∞(,)𝑆) | |
34 | 32, 33 | eleqtrrdi 2843 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
35 | 34 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) |
36 | dfss3 3970 | . 2 ⊢ (𝐴 ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) | |
37 | 35, 36 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⊆ wss 3948 class class class wbr 5148 (class class class)co 7412 supcsup 9441 ℝcr 11115 -∞cmnf 11253 ℝ*cxr 11254 < clt 11255 ≤ cle 11256 (,)cioo 13331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-ioo 13335 |
This theorem is referenced by: pimdecfgtioo 45892 pimincfltioo 45893 |
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