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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 7771 | . . . . 5 ⊢ -∞ = 𝒫 +∞ | |
2 | pnfex 7787 | . . . . . 6 ⊢ +∞ ∈ V | |
3 | 2 | pwex 4077 | . . . . 5 ⊢ 𝒫 +∞ ∈ V |
4 | 1, 3 | eqeltri 2190 | . . . 4 ⊢ -∞ ∈ V |
5 | 4 | prid2 3600 | . . 3 ⊢ -∞ ∈ {+∞, -∞} |
6 | elun2 3214 | . . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞})) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞}) |
8 | df-xr 7772 | . 2 ⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | |
9 | 7, 8 | eleqtrri 2193 | 1 ⊢ -∞ ∈ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 Vcvv 2660 ∪ cun 3039 𝒫 cpw 3480 {cpr 3498 ℝcr 7587 +∞cpnf 7765 -∞cmnf 7766 ℝ*cxr 7767 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-un 4325 ax-cnex 7679 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-pnf 7770 df-mnf 7771 df-xr 7772 |
This theorem is referenced by: elxr 9531 xrltnr 9534 mnflt 9537 mnfltpnf 9539 nltmnf 9542 mnfle 9546 xrltnsym 9547 xrlttri3 9551 ngtmnft 9568 xrrebnd 9570 xrre2 9572 xrre3 9573 ge0gtmnf 9574 xnegcl 9583 xltnegi 9586 xaddf 9595 xaddval 9596 xaddmnf1 9599 xaddmnf2 9600 pnfaddmnf 9601 mnfaddpnf 9602 xrex 9607 xltadd1 9627 xlt2add 9631 xsubge0 9632 xposdif 9633 xleaddadd 9638 elioc2 9687 elico2 9688 elicc2 9689 ioomax 9699 iccmax 9700 elioomnf 9719 unirnioo 9724 xrmaxadd 10998 reopnap 12634 blssioo 12641 tgioo 12642 |
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