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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 13414 | . 2 ⊢ (𝐵[,]𝐶) ⊆ ℝ* | |
2 | 1 | sseli 3978 | 1 ⊢ (𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 (class class class)co 7412 ℝ*cxr 11254 [,]cicc 13334 |
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-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3061 df-rex 3070 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-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-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-xr 11259 df-icc 13338 |
This theorem is referenced by: xrge0neqmnf 13436 xrge0nre 13437 isxmet2d 24152 stdbdxmet 24343 metds0 24685 metdstri 24686 metdsre 24688 metdseq0 24689 metdscnlem 24690 metnrmlem1a 24693 metnrmlem1 24694 oprpiece1res1 24795 xrge0infss 32405 xrge0mulgnn0 32622 xrge0omnd 32664 esumcvgre 33552 mblfinlem1 36988 iccintsng 44694 icoiccdif 44695 eliccnelico 44700 eliccelicod 44701 ge0xrre 44702 iblspltprt 45147 iblcncfioo 45152 itgspltprt 45153 gsumge0cl 45545 sge0tsms 45554 |
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