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Mirrors > Home > ILE Home > Th. List > xrex | 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 7804 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
2 | reex 7754 | . . 3 ⊢ ℝ ∈ V | |
3 | pnfxr 7818 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | mnfxr 7822 | . . . 4 ⊢ -∞ ∈ ℝ* | |
5 | prexg 4133 | . . . 4 ⊢ ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → {+∞, -∞} ∈ V) | |
6 | 3, 4, 5 | mp2an 422 | . . 3 ⊢ {+∞, -∞} ∈ V |
7 | 2, 6 | unex 4362 | . 2 ⊢ (ℝ ∪ {+∞, -∞}) ∈ V |
8 | 1, 7 | eqeltri 2212 | 1 ⊢ ℝ* ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 {cpr 3528 ℝcr 7619 +∞cpnf 7797 -∞cmnf 7798 ℝ*cxr 7799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-pnf 7802 df-mnf 7803 df-xr 7804 |
This theorem is referenced by: ixxval 9679 ixxf 9681 ixxex 9682 ispsmet 12492 isxmet 12514 xmetunirn 12527 blfvalps 12554 blex 12556 |
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