mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 8013

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 7994	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 8010	. . . . . 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 7995	. 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 7988 -∞cmnf 7989 ℝ^*cxr 7990

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 7993 df-mnf 7994 df-xr 7995

This theorem is referenced by: elxr 9775 xrltnr 9778 mnflt 9782 mnfltpnf 9784 nltmnf 9787 mnfle 9791 xrltnsym 9792 xrlttri3 9796 ngtmnft 9816 xrrebnd 9818 xrre2 9820 xrre3 9821 ge0gtmnf 9822 xnegcl 9831 xltnegi 9834 xaddf 9843 xaddval 9844 xaddmnf1 9847 xaddmnf2 9848 pnfaddmnf 9849 mnfaddpnf 9850 xrex 9855 xltadd1 9875 xlt2add 9879 xsubge0 9880 xposdif 9881 xleaddadd 9886 elioc2 9935 elico2 9936 elicc2 9937 ioomax 9947 iccmax 9948 elioomnf 9967 unirnioo 9972 xrmaxadd 11268 reopnap 14008 blssioo 14015 tgioo 14016

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