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Theorem xrnepnf 9844

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 9842	. . . 4 $\/$ $\/$
3		df-3or 981	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 771	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2365	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8067	. . . . 5
9		mnfnepnf 8075	. . . . . 6
10		neeq1 2377	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 717	. . . 4 $\/$
13	12	neneqd 2385	. . 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 2164

wne 2364

cr 7871

cpnf 8051

cmnf 8052

cxr 8053

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-un 4464 ax-cnex 7963 ax-resscn 7964

This theorem depends on definitions: df-bi 117 df-3or 981 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-rex 2478 df-rab 2481 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-uni 3836 df-pnf 8056 df-mnf 8057 df-xr 8058

This theorem is referenced by: xaddnepnf 9924