mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 7929

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 7910	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 7926	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4144	. . . . 5 ⊢ 𝒫 +∞ ∈ V
4	1, 3	eqeltri 2230	. . . 4 ⊢ -∞ ∈ V
5	4	prid2 3666	. . 3 ⊢ -∞ ∈ {+∞, -∞}
6		elun2 3275	. . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
7	5, 6	ax-mp 5	. 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞})
8		df-xr 7911	. 2 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
9	7, 8	eleqtrri 2233	1 ⊢ -∞ ∈ ℝ^*

Colors of variables: wff set class

Syntax hints: ∈ wcel 2128 Vcvv 2712 ∪ cun 3100 𝒫 cpw 3543 {cpr 3561 ℝcr 7726 +∞cpnf 7904 -∞cmnf 7905 ℝ^*cxr 7906

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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-un 4393 ax-cnex 7818

This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 df-pnf 7909 df-mnf 7910 df-xr 7911

This theorem is referenced by: elxr 9678 xrltnr 9681 mnflt 9685 mnfltpnf 9687 nltmnf 9690 mnfle 9694 xrltnsym 9695 xrlttri3 9699 ngtmnft 9716 xrrebnd 9718 xrre2 9720 xrre3 9721 ge0gtmnf 9722 xnegcl 9731 xltnegi 9734 xaddf 9743 xaddval 9744 xaddmnf1 9747 xaddmnf2 9748 pnfaddmnf 9749 mnfaddpnf 9750 xrex 9755 xltadd1 9775 xlt2add 9779 xsubge0 9780 xposdif 9781 xleaddadd 9786 elioc2 9835 elico2 9836 elicc2 9837 ioomax 9847 iccmax 9848 elioomnf 9867 unirnioo 9872 xrmaxadd 11153 reopnap 12925 blssioo 12932 tgioo 12933

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