xrnepnf - Intuitionistic Logic Explorer

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

Theorem xrnepnf 9218

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		pm5.61 743	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9216	. . . 4 $\/$ $\/$
3		df-3or 925	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 722	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 204	. . 3 $\/$ $\/$
6		df-ne 2256	. . 3
7	5, 6	anbi12i 448	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 7514	. . . . 5
9		mnfnepnf 7522	. . . . . 6
10		neeq1 2268	. . . . . 6
11	9, 10	mpbiri 166	. . . . 5
12	8, 11	jaoi 671	. . . 4 $\/$
13	12	neneqd 2276	. . 3 $\/$
14	13	pm4.71i 383	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 210	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 102

wb 103 $\/$ wo 664 $\/$ w3o 923

wceq 1289

wcel 1438

wne 2255

cr 7328

cpnf 7498

cmnf 7499

cxr 7500

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3949 ax-pow 4001 ax-un 4251 ax-cnex 7415 ax-resscn 7416

This theorem depends on definitions: df-bi 115 df-3or 925 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-rex 2365 df-rab 2368 df-v 2621 df-un 3001 df-in 3003 df-ss 3010 df-pw 3427 df-sn 3447 df-pr 3448 df-uni 3649 df-pnf 7503 df-mnf 7504 df-xr 7505

This theorem is referenced by: (None)