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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 11299 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
| 2 | reex 11246 | . . 3 ⊢ ℝ ∈ V | |
| 3 | prex 5437 | . . 3 ⊢ {+∞, -∞} ∈ V | |
| 4 | 2, 3 | unex 7764 | . 2 ⊢ (ℝ ∪ {+∞, -∞}) ∈ V | 
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ ℝ* ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {cpr 4628 ℝcr 11154 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 | 
| 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-xr 11299 | 
| This theorem is referenced by: ixxval 13395 ixxf 13397 ixxex 13398 limsuple 15514 limsuplt 15515 limsupbnd1 15518 prdsds 17509 letsr 18638 xrsbas 21396 xrsadd 21397 xrsmul 21398 xrsle 21400 xrs1mnd 21422 xrs10 21423 xrs1cmn 21424 xrge0subm 21425 xrge0cmn 21426 xrsds 21427 znle 21551 leordtval2 23220 lecldbas 23227 ispsmet 24314 isxmet 24334 imasdsf1olem 24383 blfvalps 24393 nmoffn 24732 nmofval 24735 xrsxmet 24831 xrge0gsumle 24855 xrge0tsms 24856 xrlimcnp 27011 xrge00 33017 xrge0tsmsd 33065 xrhval 34019 ltex 42286 leex 42287 icof 45224 elicores 45546 fuzxrpmcn 45843 gsumge0cl 46386 ovnval2b 46567 volicorescl 46568 ovnsubaddlem1 46585 | 
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