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Theorem xrnepnf 9853

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		pm5.61 795	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9851	. . . 4 $\/$ $\/$
3		df-3or 981	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 771	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2368	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8074	. . . . 5
9		mnfnepnf 8082	. . . . . 6
10		neeq1 2380	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 717	. . . 4 $\/$
13	12	neneqd 2388	. . 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 709 $\/$ w3o 979

wceq 1364

wcel 2167

wne 2367

cr 7878

cpnf 8058

cmnf 8059

cxr 8060

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-un 4468 ax-cnex 7970 ax-resscn 7971

This theorem depends on definitions: df-bi 117 df-3or 981 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-rex 2481 df-rab 2484 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-uni 3840 df-pnf 8063 df-mnf 8064 df-xr 8065

This theorem is referenced by: xaddnepnf 9933