mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 8389

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 8370	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 8386	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4322	. . . . 5 ⊢ 𝒫 +∞ ∈ V
4	1, 3	eqeltri 2311	. . . 4 ⊢ -∞ ∈ V
5	4	prid2 3818	. . 3 ⊢ -∞ ∈ {+∞, -∞}
6		elun2 3397	. . 3 ⊢ (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
7	5, 6	ax-mp 5	. 2 ⊢ -∞ ∈ (ℝ ∪ {+∞, -∞})
8		df-xr 8371	. 2 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
9	7, 8	eleqtrri 2314	1 ⊢ -∞ ∈ ℝ^*

Colors of variables: wff set class

This proof depends on syntax axioms: ∈ wcel 2209 Vcvv 2821 ∪ cun 3218 𝒫 cpw 3688 {cpr 3710 ℝcr 8185 +∞cpnf 8364 -∞cmnf 8365 ℝ^*cxr 8366

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-un 4580 ax-cnex 8277

This proof depends on definitions: df-bi 117 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-uni 3936 df-pnf 8369 df-mnf 8370 df-xr 8371

This theorem is used by: elxr 10196 xrltnr 10199 mnflt 10203 mnfltpnf 10205 nltmnf 10208 mnfle 10212 xrltnsym 10213 xrlttri3 10217 ngtmnft 10237 xrrebnd 10239 xrre2 10241 xrre3 10242 ge0gtmnf 10243 xnegcl 10252 xltnegi 10255 xaddf 10264 xaddval 10265 xaddmnf1 10268 xaddmnf2 10269 pnfaddmnf 10270 mnfaddpnf 10271 xrex 10276 xltadd1 10296 xlt2add 10300 xsubge0 10301 xposdif 10302 xleaddadd 10307 elioc2 10356 elico2 10357 elicc2 10358 ioomax 10368 iccmax 10369 elioomnf 10388 unirnioo 10393 xrmaxadd 12053 reopnap 15784 blssioo 15791 tgioo 15792 repiecelem 17292 repiecele0 17293 repiecege0 17294

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