xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 9934

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		pm5.61 796	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9933	. . . 4 $\/$ $\/$
3		df-3or 982	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 184	. . 3 $\/$ $\/$
5		df-ne 2379	. . 3
6	4, 5	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 8156	. . . . 5
8		pnfnemnf 8162	. . . . . 6
9		neeq1 2391	. . . . . 6
10	8, 9	mpbiri 168	. . . . 5
11	7, 10	jaoi 718	. . . 4 $\/$
12	11	neneqd 2399	. . 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 710 $\/$ w3o 980

wceq 1373

wcel 2178

wne 2378

cr 7959

cpnf 8139

cmnf 8140

cxr 8141

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-rab 2495 df-v 2778 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-uni 3865 df-pnf 8144 df-mnf 8145 df-xr 8146

This theorem is referenced by: xaddf 10001 xaddval 10002 xaddnemnf 10014 xaddass 10026 xlesubadd 10040 xblss2ps 14991 xblss2 14992