xrnepnf - Intuitionistic Logic Explorer

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

Theorem xrnepnf 9900

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		pm5.61 796	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9898	. . . 4 $\/$ $\/$
3		df-3or 982	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 772	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2377	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8120	. . . . 5
9		mnfnepnf 8128	. . . . . 6
10		neeq1 2389	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 718	. . . 4 $\/$
13	12	neneqd 2397	. . 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 710 $\/$ w3o 980

wceq 1373

wcel 2176

wne 2376

cr 7924

cpnf 8104

cmnf 8105

cxr 8106

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-un 4480 ax-cnex 8016 ax-resscn 8017

This theorem depends on definitions: df-bi 117 df-3or 982 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-rex 2490 df-rab 2493 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-uni 3851 df-pnf 8109 df-mnf 8110 df-xr 8111

This theorem is referenced by: xaddnepnf 9980