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Theorem xrnepnf 10159

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		pm5.61 806	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 10157	. . . 4 $\/$ $\/$
3		df-3or 1010	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 782	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2421	. . 3
7	5, 6	anbi12i 464	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8363	. . . . 5
9		mnfnepnf 8371	. . . . . 6
10		neeq1 2433	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 728	. . . 4 $\/$
13	12	neneqd 2441	. . 3 $\/$
14	13	pm4.71i 395	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 212	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104

wb 105 $\/$ wo 720 $\/$ w3o 1008

wceq 1402

wcel 2209

wne 2420

cr 8168

cpnf 8347

cmnf 8348

cxr 8349

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-un 4573 ax-cnex 8260 ax-resscn 8261

This theorem depends on definitions: df-bi 117 df-3or 1010 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-rex 2534 df-rab 2537 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-uni 3931 df-pnf 8352 df-mnf 8353 df-xr 8354

This theorem is referenced by: xaddnepnf 10239