mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 8055

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 8036	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 8052	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4208	. . . . 5 ⊢ 𝒫 +∞ ∈ V
4	1, 3	eqeltri 2262	. . . 4 ⊢ -∞ ∈ V
5	4	prid2 3721	. . 3 ⊢ -∞ ∈ {+∞, -∞}
6		elun2 3322	. . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
7	5, 6	ax-mp 5	. 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞})
8		df-xr 8037	. 2 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
9	7, 8	eleqtrri 2265	1 ⊢ -∞ ∈ ℝ^*

Colors of variables: wff set class

Syntax hints: ∈ wcel 2160 Vcvv 2756 ∪ cun 3146 𝒫 cpw 3597 {cpr 3615 ℝcr 7850 +∞cpnf 8030 -∞cmnf 8031 ℝ^*cxr 8032

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 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-un 4458 ax-cnex 7942

This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2758 df-un 3152 df-in 3154 df-ss 3161 df-pw 3599 df-sn 3620 df-pr 3621 df-uni 3832 df-pnf 8035 df-mnf 8036 df-xr 8037

This theorem is referenced by: elxr 9818 xrltnr 9821 mnflt 9825 mnfltpnf 9827 nltmnf 9830 mnfle 9834 xrltnsym 9835 xrlttri3 9839 ngtmnft 9859 xrrebnd 9861 xrre2 9863 xrre3 9864 ge0gtmnf 9865 xnegcl 9874 xltnegi 9877 xaddf 9886 xaddval 9887 xaddmnf1 9890 xaddmnf2 9891 pnfaddmnf 9892 mnfaddpnf 9893 xrex 9898 xltadd1 9918 xlt2add 9922 xsubge0 9923 xposdif 9924 xleaddadd 9929 elioc2 9978 elico2 9979 elicc2 9980 ioomax 9990 iccmax 9991 elioomnf 10010 unirnioo 10015 xrmaxadd 11316 reopnap 14559 blssioo 14566 tgioo 14567

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