xrnemnf - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrnemnf		Unicode version

Theorem xrnemnf 10109

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		pm5.61 802	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 10108	. . . 4 $\/$ $\/$
3		df-3or 1006	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 184	. . 3 $\/$ $\/$
5		df-ne 2413	. . 3
6	4, 5	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 8321	. . . . 5
8		pnfnemnf 8327	. . . . . 6
9		neeq1 2425	. . . . . 6
10	8, 9	mpbiri 168	. . . . 5
11	7, 10	jaoi 724	. . . 4 $\/$
12	11	neneqd 2433	. . 3 $\/$
13	12	pm4.71i 391	. 2 $\/$ $\/$ $/\$
14	1, 6, 13	3bitr4i 212	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104

wb 105 $\/$ wo 716 $\/$ w3o 1004

wceq 1398

wcel 2203

wne 2412

cr 8125

cpnf 8304

cmnf 8305

cxr 8306

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-un 4553 ax-setind 4658 ax-cnex 8217 ax-resscn 8218

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2814 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-uni 3914 df-pnf 8309 df-mnf 8310 df-xr 8311

This theorem is referenced by: xaddf 10176 xaddval 10177 xaddnemnf 10189 xaddass 10201 xlesubadd 10215 xblss2ps 15261 xblss2 15262