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Theorem xrnepnf 9714

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		pm5.61 784	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9712	. . . 4 $\/$ $\/$
3		df-3or 969	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 760	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 205	. . 3 $\/$ $\/$
6		df-ne 2337	. . 3
7	5, 6	anbi12i 456	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 7946	. . . . 5
9		mnfnepnf 7954	. . . . . 6
10		neeq1 2349	. . . . . 6
11	9, 10	mpbiri 167	. . . . 5
12	8, 11	jaoi 706	. . . 4 $\/$
13	12	neneqd 2357	. . 3 $\/$
14	13	pm4.71i 389	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 211	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103

wb 104 $\/$ wo 698 $\/$ w3o 967

wceq 1343

wcel 2136

wne 2336

cr 7752

cpnf 7930

cmnf 7931

cxr 7932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-un 4411 ax-cnex 7844 ax-resscn 7845

This theorem depends on definitions: df-bi 116 df-3or 969 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-rex 2450 df-rab 2453 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-uni 3790 df-pnf 7935 df-mnf 7936 df-xr 7937

This theorem is referenced by: xaddnepnf 9794