Theorem elioo4g 10336

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 10329	. . . . 5 |- ( C e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* ) )
2		elioore 10314	. . . . 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 1011	. . . 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 10330	. . 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 8371	. . . . 5 |- ( C e. RR -> C e. RR* )
9	8	3anim3i 1218	. . . 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 10312	. . 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 ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   RRcr 8178   RR*cxr 8359   < clt 8360   (,)cioo 10290

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-ioo 10294

This theorem is used by:  ivthreinc  15746