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Mirrors > Home > ILE Home > Th. List > mnfxr | GIF version |
Description: Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
mnfxr | ⊢ -∞ ∈ ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mnf 7827 | . . . . 5 ⊢ -∞ = 𝒫 +∞ | |
2 | pnfex 7843 | . . . . . 6 ⊢ +∞ ∈ V | |
3 | 2 | pwex 4115 | . . . . 5 ⊢ 𝒫 +∞ ∈ V |
4 | 1, 3 | eqeltri 2213 | . . . 4 ⊢ -∞ ∈ V |
5 | 4 | prid2 3638 | . . 3 ⊢ -∞ ∈ {+∞, -∞} |
6 | elun2 3249 | . . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞})) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞}) |
8 | df-xr 7828 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
9 | 7, 8 | eleqtrri 2216 | 1 ⊢ -∞ ∈ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 Vcvv 2689 ∪ cun 3074 𝒫 cpw 3515 {cpr 3533 ℝcr 7643 +∞cpnf 7821 -∞cmnf 7822 ℝ*cxr 7823 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-un 4363 ax-cnex 7735 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-pnf 7826 df-mnf 7827 df-xr 7828 |
This theorem is referenced by: elxr 9593 xrltnr 9596 mnflt 9599 mnfltpnf 9601 nltmnf 9604 mnfle 9608 xrltnsym 9609 xrlttri3 9613 ngtmnft 9630 xrrebnd 9632 xrre2 9634 xrre3 9635 ge0gtmnf 9636 xnegcl 9645 xltnegi 9648 xaddf 9657 xaddval 9658 xaddmnf1 9661 xaddmnf2 9662 pnfaddmnf 9663 mnfaddpnf 9664 xrex 9669 xltadd1 9689 xlt2add 9693 xsubge0 9694 xposdif 9695 xleaddadd 9700 elioc2 9749 elico2 9750 elicc2 9751 ioomax 9761 iccmax 9762 elioomnf 9781 unirnioo 9786 xrmaxadd 11062 reopnap 12746 blssioo 12753 tgioo 12754 |
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