xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 9594

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		pm5.61 784	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9593	. . . 4 $\/$ $\/$
3		df-3or 964	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 183	. . 3 $\/$ $\/$
5		df-ne 2310	. . 3
6	4, 5	anbi12i 456	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 7838	. . . . 5
8		pnfnemnf 7844	. . . . . 6
9		neeq1 2322	. . . . . 6
10	8, 9	mpbiri 167	. . . . 5
11	7, 10	jaoi 706	. . . 4 $\/$
12	11	neneqd 2330	. . 3 $\/$
13	12	pm4.71i 389	. 2 $\/$ $\/$ $/\$
14	1, 6, 13	3bitr4i 211	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103

wb 104 $\/$ wo 698 $\/$ w3o 962

wceq 1332

wcel 1481

wne 2309

cr 7643

cpnf 7821

cmnf 7822

cxr 7823

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-pnf 7826 df-mnf 7827 df-xr 7828

This theorem is referenced by: xaddf 9657 xaddval 9658 xaddnemnf 9670 xaddass 9682 xlesubadd 9696 xblss2ps 12612 xblss2 12613