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Mirrors > Home > ILE Home > Th. List > pnfxr | GIF version |
Description: Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) |
Ref | Expression |
---|---|
pnfxr | ⊢ +∞ ∈ ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3240 | . . 3 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞}) | |
2 | df-pnf 7802 | . . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ | |
3 | cnex 7744 | . . . . . . 7 ⊢ ℂ ∈ V | |
4 | 3 | uniex 4359 | . . . . . 6 ⊢ ∪ ℂ ∈ V |
5 | 4 | pwex 4107 | . . . . 5 ⊢ 𝒫 ∪ ℂ ∈ V |
6 | 2, 5 | eqeltri 2212 | . . . 4 ⊢ +∞ ∈ V |
7 | 6 | prid1 3629 | . . 3 ⊢ +∞ ∈ {+∞, -∞} |
8 | 1, 7 | sselii 3094 | . 2 ⊢ +∞ ∈ (ℝ ∪ {+∞, -∞}) |
9 | df-xr 7804 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
10 | 8, 9 | eleqtrri 2215 | 1 ⊢ +∞ ∈ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 𝒫 cpw 3510 {cpr 3528 ∪ cuni 3736 ℂcc 7618 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-un 4355 ax-cnex 7711 |
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-xr 7804 |
This theorem is referenced by: pnfex 7819 pnfnemnf 7820 xnn0xr 9045 xrltnr 9566 ltpnf 9567 mnfltpnf 9571 pnfnlt 9573 pnfge 9575 xrlttri3 9583 nltpnft 9597 xgepnf 9599 xrrebnd 9602 xrre 9603 xrre2 9604 xnegcl 9615 xaddf 9627 xaddval 9628 xaddpnf1 9629 xaddpnf2 9630 pnfaddmnf 9633 mnfaddpnf 9634 xrex 9639 xaddass2 9653 xltadd1 9659 xlt2add 9663 xsubge0 9664 xposdif 9665 xleaddadd 9670 elioc2 9719 elico2 9720 elicc2 9721 ioomax 9731 iccmax 9732 ioopos 9733 elioopnf 9750 elicopnf 9752 unirnioo 9756 elxrge0 9761 hashinfom 10524 rexico 10993 xrmaxiflemcl 11014 xrmaxadd 11030 xblpnfps 12567 xblpnf 12568 xblss2ps 12573 blssec 12607 blpnfctr 12608 reopnap 12707 blssioo 12714 |
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