Theorem elioo4g 10003

Description: Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
elioo4g	|- ( C e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ ( A < C /\ C < B ) ) )

Proof of Theorem elioo4g

Step	Hyp	Ref	Expression
1		eliooxr 9996	. . . . 5 |- ( C e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* ) )
2		elioore 9981	. . . . 5 |- ( C e. ( A (,) B ) -> C e. RR )
3	1, 2	jca 306	. . . 4 |- ( C e. ( A (,) B ) -> ( ( A e. RR* /\ B e. RR* ) /\ C e. RR ) )
4		df-3an 982	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) <-> ( ( A e. RR* /\ B e. RR* ) /\ C e. RR ) )
5	3, 4	sylibr 134	. . 3 |- ( C e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* /\ C e. RR ) )
6		eliooord 9997	. . 3 |- ( C e. ( A (,) B ) -> ( A < C /\ C < B ) )
7	5, 6	jca 306	. 2 |- ( C e. ( A (,) B ) -> ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ ( A < C /\ C < B ) ) )
8		rexr 8067	. . . . 5 |- ( C e. RR -> C e. RR* )
9	8	3anim3i 1189	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
10	9	anim1i 340	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ ( A < C /\ C < B ) ) -> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ C < B ) ) )
11		elioo3g 9979	. . 3 |- ( C e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < C /\ C < B ) ) )
12	10, 11	sylibr 134	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ ( A < C /\ C < B ) ) -> C e. ( A (,) B ) )
13	7, 12	impbii 126	1 |- ( C e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ ( A < C /\ C < B ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   RRcr 7873   RR*cxr 8055   < clt 8056   (,)cioo 9957

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-ioo 9961

This theorem is referenced by:  ivthreinc  14824