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Theorem xrnepnf 10111

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		pm5.61 802	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 10109	. . . 4 $\/$ $\/$
3		df-3or 1006	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 778	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2413	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8321	. . . . 5
9		mnfnepnf 8329	. . . . . 6
10		neeq1 2425	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 724	. . . 4 $\/$
13	12	neneqd 2433	. . 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 716 $\/$ w3o 1004

wceq 1398

wcel 2203

wne 2412

cr 8126

cpnf 8305

cmnf 8306

cxr 8307

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-un 4554 ax-cnex 8218 ax-resscn 8219

This theorem depends on definitions: df-bi 117 df-3or 1006 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-rex 2526 df-rab 2529 df-v 2815 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-uni 3915 df-pnf 8310 df-mnf 8311 df-xr 8312

This theorem is referenced by: xaddnepnf 10191