xrnemnf - Intuitionistic Logic Explorer

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Theorem xrnemnf 9557

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		pm5.61 783	. 2 $\/$ $\/$ $/\$ $\/$ $/\$
2		elxr 9556	. . . 4 $\/$ $\/$
3		df-3or 963	. . . 4 $\/$ $\/$ $\/$ $\/$
4	2, 3	bitri 183	. . 3 $\/$ $\/$
5		df-ne 2307	. . 3
6	4, 5	anbi12i 455	. 2 $/\$ $\/$ $\/$ $/\$
7		renemnf 7807	. . . . 5
8		pnfnemnf 7813	. . . . . 6
9		neeq1 2319	. . . . . 6
10	8, 9	mpbiri 167	. . . . 5
11	7, 10	jaoi 705	. . . 4 $\/$
12	11	neneqd 2327	. . 3 $\/$
13	12	pm4.71i 388	. 2 $\/$ $\/$ $/\$
14	1, 6, 13	3bitr4i 211	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3 $/\$ wa 103

wb 104 $\/$ wo 697 $\/$ w3o 961

wceq 1331

wcel 1480

wne 2306

cr 7612

cpnf 7790

cmnf 7791

cxr 7792

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705

This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-pnf 7795 df-mnf 7796 df-xr 7797

This theorem is referenced by: xaddf 9620 xaddval 9621 xaddnemnf 9633 xaddass 9645 xlesubadd 9659 xblss2ps 12562 xblss2 12563