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Mirrors > Home > MPE Home > Th. List > xrex | Structured version Visualization version GIF version |
Description: The set of extended reals exists. (Contributed by NM, 24-Dec-2006.) |
Ref | Expression |
---|---|
xrex | ⊢ ℝ* ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xr 10116 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
2 | reex 10065 | . . 3 ⊢ ℝ ∈ V | |
3 | prex 4939 | . . 3 ⊢ {+∞, -∞} ∈ V | |
4 | 2, 3 | unex 6998 | . 2 ⊢ (ℝ ∪ {+∞, -∞}) ∈ V |
5 | 1, 4 | eqeltri 2726 | 1 ⊢ ℝ* ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 Vcvv 3231 ∪ cun 3605 {cpr 4212 ℝcr 9973 +∞cpnf 10109 -∞cmnf 10110 ℝ*cxr 10111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rex 2947 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-sn 4211 df-pr 4213 df-uni 4469 df-xr 10116 |
This theorem is referenced by: ixxval 12221 ixxf 12223 ixxex 12224 limsuple 14253 limsuplt 14254 limsupbnd1 14257 prdsds 16171 letsr 17274 xrsbas 19810 xrsadd 19811 xrsmul 19812 xrsle 19814 xrs1mnd 19832 xrs10 19833 xrs1cmn 19834 xrge0subm 19835 xrge0cmn 19836 xrsds 19837 znle 19932 leordtval2 21064 lecldbas 21071 ispsmet 22156 isxmet 22176 imasdsf1olem 22225 blfvalps 22235 nmoffn 22562 nmofval 22565 xrsxmet 22659 xrge0gsumle 22683 xrge0tsms 22684 xrlimcnp 24740 xrge00 29814 xrge0tsmsd 29913 xrhval 30190 icof 39725 elicores 40078 fuzxrpmcn 40372 gsumge0cl 40906 ovnval2b 41087 volicorescl 41088 ovnsubaddlem1 41105 |
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