mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 8014

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 7995	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 8011	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4184	. . . . 5 ⊢ 𝒫 +∞ ∈ V
4	1, 3	eqeltri 2250	. . . 4 ⊢ -∞ ∈ V
5	4	prid2 3700	. . 3 ⊢ -∞ ∈ {+∞, -∞}
6		elun2 3304	. . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
7	5, 6	ax-mp 5	. 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞})
8		df-xr 7996	. 2 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
9	7, 8	eleqtrri 2253	1 ⊢ -∞ ∈ ℝ^*

Colors of variables: wff set class

Syntax hints: ∈ wcel 2148 Vcvv 2738 ∪ cun 3128 𝒫 cpw 3576 {cpr 3594 ℝcr 7810 +∞cpnf 7989 -∞cmnf 7990 ℝ^*cxr 7991

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 4122 ax-pow 4175 ax-un 4434 ax-cnex 7902

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 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-uni 3811 df-pnf 7994 df-mnf 7995 df-xr 7996

This theorem is referenced by: elxr 9776 xrltnr 9779 mnflt 9783 mnfltpnf 9785 nltmnf 9788 mnfle 9792 xrltnsym 9793 xrlttri3 9797 ngtmnft 9817 xrrebnd 9819 xrre2 9821 xrre3 9822 ge0gtmnf 9823 xnegcl 9832 xltnegi 9835 xaddf 9844 xaddval 9845 xaddmnf1 9848 xaddmnf2 9849 pnfaddmnf 9850 mnfaddpnf 9851 xrex 9856 xltadd1 9876 xlt2add 9880 xsubge0 9881 xposdif 9882 xleaddadd 9887 elioc2 9936 elico2 9937 elicc2 9938 ioomax 9948 iccmax 9949 elioomnf 9968 unirnioo 9973 xrmaxadd 11269 reopnap 14041 blssioo 14048 tgioo 14049

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