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 9975 | . . . . . 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 9962 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
6 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
7 | 4, 6 | sstrd 3578 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
9 | 8 | supxrcld 38321 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | 3, 9 | syl5eqel 2692 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℝ*) |
11 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
12 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sseldd 3569 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
14 | 13 | mnfltd 11834 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ < 𝑥) |
15 | 7 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
16 | supxrub 12026 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) | |
17 | 8, 12, 16 | syl2anc 691 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) |
18 | 3 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = sup(𝐴, ℝ*, < )) |
19 | 18 | eqcomd 2616 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) = 𝑆) |
20 | 17, 19 | breqtrd 4609 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ 𝑆) |
21 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆) | |
22 | 21 | eqcomd 2616 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑆 → 𝑆 = 𝑥) |
23 | 22 | adantl 481 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 = 𝑥) |
24 | simpl 472 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑥 ∈ 𝐴) | |
25 | 23, 24 | eqeltrd 2688 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
26 | 25 | adantll 746 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
27 | ressioosup.n | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) | |
28 | 27 | ad2antrr 758 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → ¬ 𝑆 ∈ 𝐴) |
29 | 26, 28 | pm2.65da 598 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝑆) |
30 | 29 | neqned 2789 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝑆) |
31 | 15, 10, 20, 30 | xrleneltd 38480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 < 𝑆) |
32 | 2, 10, 13, 14, 31 | eliood 38567 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (-∞(,)𝑆)) |
33 | ressioosup.i | . . . 4 ⊢ 𝐼 = (-∞(,)𝑆) | |
34 | 32, 33 | syl6eleqr 2699 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
35 | 34 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) |
36 | dfss3 3558 | . 2 ⊢ (𝐴 ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) | |
37 | 35, 36 | sylibr 223 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 class class class wbr 4583 (class class class)co 6549 supcsup 8229 ℝcr 9814 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 |
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 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 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-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-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-ioo 12050 |
This theorem is referenced by: pimdecfgtioo 39604 pimincfltioo 39605 |
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