mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 8076

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 8057	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 8073	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4212	. . . . 5 ⊢ 𝒫 +∞ ∈ V
4	1, 3	eqeltri 2266	. . . 4 ⊢ -∞ ∈ V
5	4	prid2 3725	. . 3 ⊢ -∞ ∈ {+∞, -∞}
6		elun2 3327	. . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
7	5, 6	ax-mp 5	. 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞})
8		df-xr 8058	. 2 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
9	7, 8	eleqtrri 2269	1 ⊢ -∞ ∈ ℝ^*

Colors of variables: wff set class

Syntax hints: ∈ wcel 2164 Vcvv 2760 ∪ cun 3151 𝒫 cpw 3601 {cpr 3619 ℝcr 7871 +∞cpnf 8051 -∞cmnf 8052 ℝ^*cxr 8053

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-un 4464 ax-cnex 7963

This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-uni 3836 df-pnf 8056 df-mnf 8057 df-xr 8058

This theorem is referenced by: elxr 9842 xrltnr 9845 mnflt 9849 mnfltpnf 9851 nltmnf 9854 mnfle 9858 xrltnsym 9859 xrlttri3 9863 ngtmnft 9883 xrrebnd 9885 xrre2 9887 xrre3 9888 ge0gtmnf 9889 xnegcl 9898 xltnegi 9901 xaddf 9910 xaddval 9911 xaddmnf1 9914 xaddmnf2 9915 pnfaddmnf 9916 mnfaddpnf 9917 xrex 9922 xltadd1 9942 xlt2add 9946 xsubge0 9947 xposdif 9948 xleaddadd 9953 elioc2 10002 elico2 10003 elicc2 10004 ioomax 10014 iccmax 10015 elioomnf 10034 unirnioo 10039 xrmaxadd 11404 reopnap 14706 blssioo 14713 tgioo 14714

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