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Theorem xrnepnf 9452

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		pm5.61 766	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9450	. . . 4 $\/$ $\/$
3		df-3or 944	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 742	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 205	. . 3 $\/$ $\/$
6		df-ne 2281	. . 3
7	5, 6	anbi12i 453	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 7731	. . . . 5
9		mnfnepnf 7739	. . . . . 6
10		neeq1 2293	. . . . . 6
11	9, 10	mpbiri 167	. . . . 5
12	8, 11	jaoi 688	. . . 4 $\/$
13	12	neneqd 2301	. . 3 $\/$
14	13	pm4.71i 386	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 211	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103

wb 104 $\/$ wo 680 $\/$ w3o 942

wceq 1312

wcel 1461

wne 2280

cr 7540

cpnf 7715

cmnf 7716

cxr 7717

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-un 4313 ax-cnex 7630 ax-resscn 7631

This theorem depends on definitions: df-bi 116 df-3or 944 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-rex 2394 df-rab 2397 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-pnf 7720 df-mnf 7721 df-xr 7722

This theorem is referenced by: xaddnepnf 9528