xrnepnf - Metamath Proof Explorer

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Theorem xrnepnf 11952

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		pm5.61 749	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 11950	. . . 4 $\/$ $\/$
3		df-3or 1038	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 549	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 286	. . 3 $\/$ $\/$
6		df-ne 2795	. . 3
7	5, 6	anbi12i 733	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 10087	. . . . 5
9		mnfnepnf 10095	. . . . . 6
10		neeq1 2856	. . . . . 6
11	9, 10	mpbiri 248	. . . . 5
12	8, 11	jaoi 394	. . . 4 $\/$
13	12	neneqd 2799	. . 3 $\/$
14	13	pm4.71i 664	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 292	1 $/\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wb 196 $\/$ wo 383 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

wne 2794

cr 9935

cpnf 10071

cmnf 10072

cxr 10073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 df-pnf 10076 df-mnf 10077 df-xr 10078

This theorem is referenced by: xaddnepnf 12068 xlt2addrd 29523 xrlexaddrp 39568 xrnpnfmnf 39705