mnfxr - Intuitionistic Logic Explorer

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Theorem mnfxr 8382

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 8363	. . . . 5 ⊢ -∞ = 𝒫 +∞
2		pnfex 8379	. . . . . 6 ⊢ +∞ ∈ V
3	2	pwex 4320	. . . . 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 8364	. 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 8178 +∞cpnf 8357 -∞cmnf 8358 ℝ^*cxr 8359

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 4249 ax-pow 4311 ax-un 4578 ax-cnex 8270

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 8362 df-mnf 8363 df-xr 8364

This theorem is used by: elxr 10180 xrltnr 10183 mnflt 10187 mnfltpnf 10189 nltmnf 10192 mnfle 10196 xrltnsym 10197 xrlttri3 10201 ngtmnft 10221 xrrebnd 10223 xrre2 10225 xrre3 10226 ge0gtmnf 10227 xnegcl 10236 xltnegi 10239 xaddf 10248 xaddval 10249 xaddmnf1 10252 xaddmnf2 10253 pnfaddmnf 10254 mnfaddpnf 10255 xrex 10260 xltadd1 10280 xlt2add 10284 xsubge0 10285 xposdif 10286 xleaddadd 10291 elioc2 10340 elico2 10341 elicc2 10342 ioomax 10352 iccmax 10353 elioomnf 10372 unirnioo 10377 xrmaxadd 12029 reopnap 15649 blssioo 15656 tgioo 15657 repiecelem 17086 repiecele0 17087 repiecege0 17088

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