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Theorem xrnepnf 8700

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		pm5.61 708	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 8696	. . . 4 $\/$ $\/$
3		df-3or 886	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 687	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 195	. . 3 $\/$ $\/$
6		df-ne 2206	. . 3
7	5, 6	anbi12i 433	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 7073	. . . . 5
9		mnfnepnf 8698	. . . . . 6
10		neeq1 2218	. . . . . 6
11	9, 10	mpbiri 157	. . . . 5
12	8, 11	jaoi 636	. . . 4 $\/$
13	12	neneqd 2226	. . 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 629 $\/$ w3o 884

wceq 1243

wcel 1393

wne 2204

cr 6888

cpnf 7057

cmnf 7058

cxr 7059

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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-un 4170 ax-cnex 6975 ax-resscn 6976

This theorem depends on definitions: df-bi 110 df-3or 886 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-rex 2312 df-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-mnf 7063 df-xr 7064

This theorem is referenced by: (None)