xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 10129

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 10128	. . . 4 $\/$ $\/$
3		df-3or 1006	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 184	. . 3 $\/$ $\/$
5		df-ne 2415	. . 3
6	4, 5	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 8338	. . . . 5
8		pnfnemnf 8344	. . . . . 6
9		neeq1 2427	. . . . . 6
10	8, 9	mpbiri 168	. . . . 5
11	7, 10	jaoi 724	. . . 4 $\/$
12	11	neneqd 2435	. . 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 2205

wne 2414

cr 8142

cpnf 8321

cmnf 8322

cxr 8323

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235

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 1812 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-uni 3920 df-pnf 8326 df-mnf 8327 df-xr 8328

This theorem is referenced by: xaddf 10196 xaddval 10197 xaddnemnf 10209 xaddass 10221 xlesubadd 10235 xblss2ps 15395 xblss2 15396