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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 3101 | . . 3 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞}) | |
2 | df-pnf 6859 | . . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ | |
3 | cnex 6803 | . . . . . . 7 ⊢ ℂ ∈ V | |
4 | 3 | uniex 4140 | . . . . . 6 ⊢ ∪ ℂ ∈ V |
5 | 4 | pwex 3923 | . . . . 5 ⊢ 𝒫 ∪ ℂ ∈ V |
6 | 2, 5 | eqeltri 2107 | . . . 4 ⊢ +∞ ∈ V |
7 | 6 | prid1 3467 | . . 3 ⊢ +∞ ∈ {+∞, -∞} |
8 | 1, 7 | sselii 2936 | . 2 ⊢ +∞ ∈ (ℝ ∪ {+∞, -∞}) |
9 | df-xr 6861 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
10 | 8, 9 | eleqtrri 2110 | 1 ⊢ +∞ ∈ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1390 Vcvv 2551 ∪ cun 2909 𝒫 cpw 3351 {cpr 3368 ∪ cuni 3571 ℂcc 6709 ℝcr 6710 +∞cpnf 6854 -∞cmnf 6855 ℝ*cxr 6856 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-un 4136 ax-cnex 6774 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-pnf 6859 df-xr 6861 |
This theorem is referenced by: pnfex 8463 pnfnemnf 8467 xrltnr 8471 ltpnf 8472 mnfltpnf 8476 pnfnlt 8478 pnfge 8480 xrlttri3 8488 nltpnft 8500 xrrebnd 8502 xrre 8503 xrre2 8504 xnegcl 8515 xrex 8526 elioc2 8575 elico2 8576 elicc2 8577 ioomax 8587 iccmax 8588 ioopos 8589 elioopnf 8606 elicopnf 8608 unirnioo 8612 elxrge0 8617 |
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