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Theorem xrnepnf 9920

Description: An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xrnepnf	$/\$ $\/$

**Proof of Theorem xrnepnf**
Step	Hyp	Ref	Expression
1		pm5.61 796	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9918	. . . 4 $\/$ $\/$
3		df-3or 982	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 772	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 206	. . 3 $\/$ $\/$
6		df-ne 2378	. . 3
7	5, 6	anbi12i 460	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 8140	. . . . 5
9		mnfnepnf 8148	. . . . . 6
10		neeq1 2390	. . . . . 6
11	9, 10	mpbiri 168	. . . . 5
12	8, 11	jaoi 718	. . . 4 $\/$
13	12	neneqd 2398	. . 3 $\/$
14	13	pm4.71i 391	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 212	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 104

wb 105 $\/$ wo 710 $\/$ w3o 980

wceq 1373

wcel 2177

wne 2377

cr 7944

cpnf 8124

cmnf 8125

cxr 8126

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 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-un 4488 ax-cnex 8036 ax-resscn 8037

This theorem depends on definitions: df-bi 117 df-3or 982 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-rex 2491 df-rab 2494 df-v 2775 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-uni 3857 df-pnf 8129 df-mnf 8130 df-xr 8131

This theorem is referenced by: xaddnepnf 10000