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Theorem xrnepnf 9735

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		pm5.61 789	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9733	. . . 4 $\/$ $\/$
3		df-3or 974	. . . 4 $\/$ $\/$ $\/$ $\/$
4		or32 765	. . . 4 $\/$ $\/$ $\/$ $\/$
5	2, 3, 4	3bitri 205	. . 3 $\/$ $\/$
6		df-ne 2341	. . 3
7	5, 6	anbi12i 457	. 2 $/\$ $\/$ $\/$ $/\$
8		renepnf 7967	. . . . 5
9		mnfnepnf 7975	. . . . . 6
10		neeq1 2353	. . . . . 6
11	9, 10	mpbiri 167	. . . . 5
12	8, 11	jaoi 711	. . . 4 $\/$
13	12	neneqd 2361	. . 3 $\/$
14	13	pm4.71i 389	. 2 $\/$ $\/$ $/\$
15	1, 7, 14	3bitr4i 211	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103

wb 104 $\/$ wo 703 $\/$ w3o 972

wceq 1348

wcel 2141

wne 2340

cr 7773

cpnf 7951

cmnf 7952

cxr 7953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-un 4418 ax-cnex 7865 ax-resscn 7866

This theorem depends on definitions: df-bi 116 df-3or 974 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-pnf 7956 df-mnf 7957 df-xr 7958

This theorem is referenced by: xaddnepnf 9815