xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 9794

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		pm5.61 795	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9793	. . . 4 $\/$ $\/$
3		df-3or 980	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 184	. . 3 $\/$ $\/$
5		df-ne 2360	. . 3
6	4, 5	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 8023	. . . . 5
8		pnfnemnf 8029	. . . . . 6
9		neeq1 2372	. . . . . 6
10	8, 9	mpbiri 168	. . . . 5
11	7, 10	jaoi 717	. . . 4 $\/$
12	11	neneqd 2380	. . 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 709 $\/$ w3o 978

wceq 1363

wcel 2159

wne 2359

cr 7827

cpnf 8006

cmnf 8007

cxr 8008

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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920

This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-uni 3824 df-pnf 8011 df-mnf 8012 df-xr 8013

This theorem is referenced by: xaddf 9861 xaddval 9862 xaddnemnf 9874 xaddass 9886 xlesubadd 9900 xblss2ps 14287 xblss2 14288