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| Mirrors > Home > MPE Home > Th. List > eliccxr | Structured version Visualization version GIF version | ||
| Description: A member of a closed interval is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| eliccxr | ⊢ (𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iccssxr 13470 | . 2 ⊢ (𝐵[,]𝐶) ⊆ ℝ* | |
| 2 | 1 | sseli 3979 | 1 ⊢ (𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 ℝ*cxr 11294 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-xr 11299 df-icc 13394 |
| This theorem is referenced by: xrge0neqmnf 13492 xrge0nre 13493 isxmet2d 24337 stdbdxmet 24528 metds0 24872 metdstri 24873 metdsre 24875 metdseq0 24876 metdscnlem 24877 metnrmlem1a 24880 metnrmlem1 24881 oprpiece1res1 24982 xrge0infss 32764 xrge0mulgnn0 33020 xrge0omnd 33088 esumcvgre 34092 mblfinlem1 37664 iccintsng 45536 icoiccdif 45537 eliccnelico 45542 eliccelicod 45543 ge0xrre 45544 iblspltprt 45988 iblcncfioo 45993 itgspltprt 45994 gsumge0cl 46386 sge0tsms 46395 |
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