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Theorem xrnepnf 10012

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		pm5.61 801	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 10010	. . . 4 $\/$ $\/$
3		df-3or 1005	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 777	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2403	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8226	. . . . 5
9		mnfnepnf 8234	. . . . . 6
10		neeq1 2415	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 723	. . . 4 $\/$
13	12	neneqd 2423	. . 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 715 $\/$ w3o 1003

wceq 1397

wcel 2202

wne 2402

cr 8030

cpnf 8210

cmnf 8211

cxr 8212

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-un 4530 ax-cnex 8122 ax-resscn 8123

This theorem depends on definitions: df-bi 117 df-3or 1005 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-rex 2516 df-rab 2519 df-v 2804 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-uni 3894 df-pnf 8215 df-mnf 8216 df-xr 8217

This theorem is referenced by: xaddnepnf 10092