xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 9846

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 9845	. . . 4 $\/$ $\/$
3		df-3or 981	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 184	. . 3 $\/$ $\/$
5		df-ne 2365	. . 3
6	4, 5	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 8070	. . . . 5
8		pnfnemnf 8076	. . . . . 6
9		neeq1 2377	. . . . . 6
10	8, 9	mpbiri 168	. . . . 5
11	7, 10	jaoi 717	. . . 4 $\/$
12	11	neneqd 2385	. . 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 979

wceq 1364

wcel 2164

wne 2364

cr 7873

cpnf 8053

cmnf 8054

cxr 8055

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-uni 3837 df-pnf 8058 df-mnf 8059 df-xr 8060

This theorem is referenced by: xaddf 9913 xaddval 9914 xaddnemnf 9926 xaddass 9938 xlesubadd 9952 xblss2ps 14583 xblss2 14584