xrnemnf - Metamath Proof Explorer

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Theorem xrnemnf 11951

Description: An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xrnemnf	$/\$ $\/$

**Proof of Theorem xrnemnf**
Step	Hyp	Ref	Expression
1		pm5.61 749	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 11950	. . . 4 $\/$ $\/$
3		df-3or 1038	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 264	. . 3 $\/$ $\/$
5		df-ne 2795	. . 3
6	4, 5	anbi12i 733	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 10088	. . . . 5
8		pnfnemnf 10094	. . . . . 6
9		neeq1 2856	. . . . . 6
10	8, 9	mpbiri 248	. . . . 5
11	7, 10	jaoi 394	. . . 4 $\/$
12	11	neneqd 2799	. . 3 $\/$
13	12	pm4.71i 664	. 2 $\/$ $\/$ $/\$
14	1, 6, 13	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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078

This theorem is referenced by: xaddnemnf 12067 xaddass 12079 xlesubadd 12093 xrge0nre 12277 xblss2ps 22206 xblss2 22207 nmoix 22533 nmoleub 22535 blcvx 22601 xrge0tsms 22637 metdstri 22654 nmoleub2lem 22914 xrge0tsmsd 29785 esumcvgre 30153 icorempt2 33199 xrnmnfpnf 39256 xrred 39581