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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 3281 | . . 3 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞}) | |
2 | df-pnf 7926 | . . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ | |
3 | cnex 7868 | . . . . . . 7 ⊢ ℂ ∈ V | |
4 | 3 | uniex 4409 | . . . . . 6 ⊢ ∪ ℂ ∈ V |
5 | 4 | pwex 4156 | . . . . 5 ⊢ 𝒫 ∪ ℂ ∈ V |
6 | 2, 5 | eqeltri 2237 | . . . 4 ⊢ +∞ ∈ V |
7 | 6 | prid1 3676 | . . 3 ⊢ +∞ ∈ {+∞, -∞} |
8 | 1, 7 | sselii 3134 | . 2 ⊢ +∞ ∈ (ℝ ∪ {+∞, -∞}) |
9 | df-xr 7928 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
10 | 8, 9 | eleqtrri 2240 | 1 ⊢ +∞ ∈ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 𝒫 cpw 3553 {cpr 3571 ∪ cuni 3783 ℂcc 7742 ℝcr 7743 +∞cpnf 7921 -∞cmnf 7922 ℝ*cxr 7923 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 ax-cnex 7835 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-pnf 7926 df-xr 7928 |
This theorem is referenced by: pnfex 7943 pnfnemnf 7944 xnn0xr 9173 xrltnr 9706 ltpnf 9707 mnfltpnf 9712 pnfnlt 9714 pnfge 9716 xrlttri3 9724 xnn0dcle 9729 nltpnft 9741 xgepnf 9743 xrrebnd 9746 xrre 9747 xrre2 9748 xnegcl 9759 xaddf 9771 xaddval 9772 xaddpnf1 9773 xaddpnf2 9774 pnfaddmnf 9777 mnfaddpnf 9778 xrex 9783 xaddass2 9797 xltadd1 9803 xlt2add 9807 xsubge0 9808 xposdif 9809 xleaddadd 9814 elioc2 9863 elico2 9864 elicc2 9865 ioomax 9875 iccmax 9876 ioopos 9877 elioopnf 9894 elicopnf 9896 unirnioo 9900 elxrge0 9905 dfrp2 10189 elicore 10192 hashinfom 10680 rexico 11149 xrmaxiflemcl 11172 xrmaxadd 11188 fprodge0 11564 fprodge1 11566 pcxcl 12220 pc2dvds 12238 pcadd 12248 xblpnfps 12939 xblpnf 12940 xblss2ps 12945 blssec 12979 blpnfctr 12980 reopnap 13079 blssioo 13086 |
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