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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 3107 | . . 3 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞}) | |
2 | df-pnf 7062 | . . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ | |
3 | cnex 7005 | . . . . . . 7 ⊢ ℂ ∈ V | |
4 | 3 | uniex 4174 | . . . . . 6 ⊢ ∪ ℂ ∈ V |
5 | 4 | pwex 3932 | . . . . 5 ⊢ 𝒫 ∪ ℂ ∈ V |
6 | 2, 5 | eqeltri 2110 | . . . 4 ⊢ +∞ ∈ V |
7 | 6 | prid1 3476 | . . 3 ⊢ +∞ ∈ {+∞, -∞} |
8 | 1, 7 | sselii 2942 | . 2 ⊢ +∞ ∈ (ℝ ∪ {+∞, -∞}) |
9 | df-xr 7064 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
10 | 8, 9 | eleqtrri 2113 | 1 ⊢ +∞ ∈ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 Vcvv 2557 ∪ cun 2915 𝒫 cpw 3359 {cpr 3376 ∪ cuni 3580 ℂcc 6887 ℝcr 6888 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-un 4170 ax-cnex 6975 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-xr 7064 |
This theorem is referenced by: pnfex 8693 pnfnemnf 8697 xrltnr 8701 ltpnf 8702 mnfltpnf 8706 pnfnlt 8708 pnfge 8710 xrlttri3 8718 nltpnft 8730 xrrebnd 8732 xrre 8733 xrre2 8734 xnegcl 8745 xrex 8756 elioc2 8805 elico2 8806 elicc2 8807 ioomax 8817 iccmax 8818 ioopos 8819 elioopnf 8836 elicopnf 8838 unirnioo 8842 elxrge0 8847 |
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