xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 8853

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		pm5.61 740	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 8850	. . . 4 $\/$ $\/$
3		df-3or 920	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 182	. . 3 $\/$ $\/$
5		df-ne 2246	. . 3
6	4, 5	anbi12i 447	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 7167	. . . . 5
8		pnfnemnf 8851	. . . . . 6
9		neeq1 2258	. . . . . 6
10	8, 9	mpbiri 166	. . . . 5
11	7, 10	jaoi 668	. . . 4 $\/$
12	11	neneqd 2266	. . 3 $\/$
13	12	pm4.71i 383	. 2 $\/$ $\/$ $/\$
14	1, 6, 13	3bitr4i 210	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 102

wb 103 $\/$ wo 661 $\/$ w3o 918

wceq 1284

wcel 1433

wne 2245

cr 6980

cpnf 7150

cmnf 7151

cxr 7152

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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-pnf 7155 df-mnf 7156 df-xr 7157

This theorem is referenced by: (None)