xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 9852

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 9851	. . . 4 $\/$ $\/$
3		df-3or 981	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 184	. . 3 $\/$ $\/$
5		df-ne 2368	. . 3
6	4, 5	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 8075	. . . . 5
8		pnfnemnf 8081	. . . . . 6
9		neeq1 2380	. . . . . 6
10	8, 9	mpbiri 168	. . . . 5
11	7, 10	jaoi 717	. . . 4 $\/$
12	11	neneqd 2388	. . 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 2167

wne 2367

cr 7878

cpnf 8058

cmnf 8059

cxr 8060

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-uni 3840 df-pnf 8063 df-mnf 8064 df-xr 8065

This theorem is referenced by: xaddf 9919 xaddval 9920 xaddnemnf 9932 xaddass 9944 xlesubadd 9958 xblss2ps 14640 xblss2 14641