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Theorem xrnepnf 8470

Description: An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xrnepnf	$/\$ $\/$

**Proof of Theorem xrnepnf**
Step	Hyp	Ref	Expression
1		pm5.61 707	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 8466	. . . 4 $\/$ $\/$
3		df-3or 885	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 686	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 195	. . 3 $\/$ $\/$
6		df-ne 2203	. . 3
7	5, 6	anbi12i 433	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 6870	. . . . 5
9		mnfnepnf 8468	. . . . . 6
10		neeq1 2213	. . . . . 6
11	9, 10	mpbiri 157	. . . . 5
12	8, 11	jaoi 635	. . . 4 $\/$
13	12	neneqd 2221	. . 3 $\/$
14	13	pm4.71i 371	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 201	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 97

wb 98 $\/$ wo 628 $\/$ w3o 883

wceq 1242

wcel 1390

wne 2201

cr 6710

cpnf 6854

cmnf 6855

cxr 6856

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-un 4136 ax-cnex 6774 ax-resscn 6775

This theorem depends on definitions: df-bi 110 df-3or 885 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-pnf 6859 df-mnf 6860 df-xr 6861

This theorem is referenced by: (None)