xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 10073

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 10072	. . . 4 $\/$ $\/$
3		df-3or 1006	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 184	. . 3 $\/$ $\/$
5		df-ne 2404	. . 3
6	4, 5	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 8287	. . . . 5
8		pnfnemnf 8293	. . . . . 6
9		neeq1 2416	. . . . . 6
10	8, 9	mpbiri 168	. . . . 5
11	7, 10	jaoi 724	. . . 4 $\/$
12	11	neneqd 2424	. . 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 2202

wne 2403

cr 8091

cpnf 8270

cmnf 8271

cxr 8272

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 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184

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 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-uni 3899 df-pnf 8275 df-mnf 8276 df-xr 8277

This theorem is referenced by: xaddf 10140 xaddval 10141 xaddnemnf 10153 xaddass 10165 xlesubadd 10179 xblss2ps 15215 xblss2 15216