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Theorem xrnepnf 9974

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		pm5.61 799	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9972	. . . 4 $\/$ $\/$
3		df-3or 1003	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 775	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2401	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8194	. . . . 5
9		mnfnepnf 8202	. . . . . 6
10		neeq1 2413	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 721	. . . 4 $\/$
13	12	neneqd 2421	. . 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 713 $\/$ w3o 1001

wceq 1395

wcel 2200

wne 2400

cr 7998

cpnf 8178

cmnf 8179

cxr 8180

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-un 4524 ax-cnex 8090 ax-resscn 8091

This theorem depends on definitions: df-bi 117 df-3or 1003 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-rex 2514 df-rab 2517 df-v 2801 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-uni 3889 df-pnf 8183 df-mnf 8184 df-xr 8185

This theorem is referenced by: xaddnepnf 10054