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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 12811 | . 2 ⊢ (𝐵[,]𝐶) ⊆ ℝ* | |
2 | 1 | sseli 3961 | 1 ⊢ (𝐴 ∈ (𝐵[,]𝐶) → 𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7148 ℝ*cxr 10666 [,]cicc 12733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-xr 10671 df-icc 12737 |
This theorem is referenced by: xrge0neqmnf 12832 xrge0nre 12833 isxmet2d 22929 stdbdxmet 23117 metds0 23450 metdstri 23451 metdsre 23453 metdseq0 23454 metdscnlem 23455 metnrmlem1a 23458 metnrmlem1 23459 oprpiece1res1 23547 xrge0infss 30476 xrge0mulgnn0 30669 xrge0omnd 30705 esumcvgre 31343 mblfinlem1 34921 iccintsng 41789 icoiccdif 41790 eliccnelico 41795 eliccelicod 41796 ge0xrre 41797 iblspltprt 42248 iblcncfioo 42253 itgspltprt 42254 gsumge0cl 42644 sge0tsms 42653 |
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