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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 12808 | . 2 ⊢ (𝐵[,]𝐶) ⊆ ℝ* | |
2 | 1 | sseli 3911 | 1 ⊢ (𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 (class class class)co 7135 ℝ*cxr 10663 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-xr 10668 df-icc 12733 |
This theorem is referenced by: xrge0neqmnf 12830 xrge0nre 12831 isxmet2d 22934 stdbdxmet 23122 metds0 23455 metdstri 23456 metdsre 23458 metdseq0 23459 metdscnlem 23460 metnrmlem1a 23463 metnrmlem1 23464 oprpiece1res1 23556 xrge0infss 30510 xrge0mulgnn0 30723 xrge0omnd 30762 esumcvgre 31460 mblfinlem1 35094 iccintsng 42160 icoiccdif 42161 eliccnelico 42166 eliccelicod 42167 ge0xrre 42168 iblspltprt 42615 iblcncfioo 42620 itgspltprt 42621 gsumge0cl 43010 sge0tsms 43019 |
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