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Theorem xrnepnf 9705

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 9703	. . . 4 $\/$ $\/$
3		df-3or 968	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 760	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 205	. . 3 $\/$ $\/$
6		df-ne 2335	. . 3
7	5, 6	anbi12i 456	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 7937	. . . . 5
9		mnfnepnf 7945	. . . . . 6
10		neeq1 2347	. . . . . 6
11	9, 10	mpbiri 167	. . . . 5
12	8, 11	jaoi 706	. . . 4 $\/$
13	12	neneqd 2355	. . 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 966

wceq 1342

wcel 2135

wne 2334

cr 7743

cpnf 7921

cmnf 7922

cxr 7923

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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 ax-cnex 7835 ax-resscn 7836

This theorem depends on definitions: df-bi 116 df-3or 968 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-pnf 7926 df-mnf 7927 df-xr 7928

This theorem is referenced by: xaddnepnf 9785