mnfxr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mnfxr		GIF version

Theorem mnfxr 7955

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 7936	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 7952	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4162	. . . . 5 ⊢ 𝒫 +∞ ∈ V
4	1, 3	eqeltri 2239	. . . 4 ⊢ -∞ ∈ V
5	4	prid2 3683	. . 3 ⊢ -∞ ∈ {+∞, -∞}
6		elun2 3290	. . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
7	5, 6	ax-mp 5	. 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞})
8		df-xr 7937	. 2 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
9	7, 8	eleqtrri 2242	1 ⊢ -∞ ∈ ℝ^*

Colors of variables: wff set class

Syntax hints: ∈ wcel 2136 Vcvv 2726 ∪ cun 3114 𝒫 cpw 3559 {cpr 3577 ℝcr 7752 +∞cpnf 7930 -∞cmnf 7931 ℝ^*cxr 7932

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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-un 4411 ax-cnex 7844

This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-uni 3790 df-pnf 7935 df-mnf 7936 df-xr 7937

This theorem is referenced by: elxr 9712 xrltnr 9715 mnflt 9719 mnfltpnf 9721 nltmnf 9724 mnfle 9728 xrltnsym 9729 xrlttri3 9733 ngtmnft 9753 xrrebnd 9755 xrre2 9757 xrre3 9758 ge0gtmnf 9759 xnegcl 9768 xltnegi 9771 xaddf 9780 xaddval 9781 xaddmnf1 9784 xaddmnf2 9785 pnfaddmnf 9786 mnfaddpnf 9787 xrex 9792 xltadd1 9812 xlt2add 9816 xsubge0 9817 xposdif 9818 xleaddadd 9823 elioc2 9872 elico2 9873 elicc2 9874 ioomax 9884 iccmax 9885 elioomnf 9904 unirnioo 9909 xrmaxadd 11202 reopnap 13178 blssioo 13185 tgioo 13186

W3C validator