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Theorem xrnepnf 9810

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		pm5.61 795	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9808	. . . 4 $\/$ $\/$
3		df-3or 981	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 771	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2361	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8036	. . . . 5
9		mnfnepnf 8044	. . . . . 6
10		neeq1 2373	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 717	. . . 4 $\/$
13	12	neneqd 2381	. . 3 $\/$
14	13	pm4.71i 391	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 212	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104

wb 105 $\/$ wo 709 $\/$ w3o 979

wceq 1364

wcel 2160

wne 2360

cr 7841

cpnf 8020

cmnf 8021

cxr 8022

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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-un 4451 ax-cnex 7933 ax-resscn 7934

This theorem depends on definitions: df-bi 117 df-3or 981 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-pnf 8025 df-mnf 8026 df-xr 8027

This theorem is referenced by: xaddnepnf 9890