mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 8012

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.)

**Assertion**
Ref	Expression
mnfxr	⊢ -∞ ∈ ℝ^*

**Proof of Theorem mnfxr**
Step	Hyp	Ref	Expression
1		df-mnf 7993	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 8009	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4183	. . . . 5 ⊢ 𝒫 +∞ ∈ V
4	1, 3	eqeltri 2250	. . . 4 ⊢ -∞ ∈ V
5	4	prid2 3699	. . 3 ⊢ -∞ ∈ {+∞, -∞}
6		elun2 3303	. . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
7	5, 6	ax-mp 5	. 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞})
8		df-xr 7994	. 2 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
9	7, 8	eleqtrri 2253	1 ⊢ -∞ ∈ ℝ^*

Colors of variables: wff set class

Syntax hints: ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 𝒫 cpw 3575 {cpr 3593 ℝcr 7809 +∞cpnf 7987 -∞cmnf 7988 ℝ^*cxr 7989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-un 4433 ax-cnex 7901

This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-pnf 7992 df-mnf 7993 df-xr 7994

This theorem is referenced by: elxr 9774 xrltnr 9777 mnflt 9781 mnfltpnf 9783 nltmnf 9786 mnfle 9790 xrltnsym 9791 xrlttri3 9795 ngtmnft 9815 xrrebnd 9817 xrre2 9819 xrre3 9820 ge0gtmnf 9821 xnegcl 9830 xltnegi 9833 xaddf 9842 xaddval 9843 xaddmnf1 9846 xaddmnf2 9847 pnfaddmnf 9848 mnfaddpnf 9849 xrex 9854 xltadd1 9874 xlt2add 9878 xsubge0 9879 xposdif 9880 xleaddadd 9885 elioc2 9934 elico2 9935 elicc2 9936 ioomax 9946 iccmax 9947 elioomnf 9966 unirnioo 9971 xrmaxadd 11264 reopnap 13931 blssioo 13938 tgioo 13939

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