Theorem elioo4g 10230

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 10223	. . . . 5 |- ( C e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* ) )
2		elioore 10208	. . . . 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 1007	. . . 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 10224	. . 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 8284	. . . . 5 |- ( C e. RR -> C e. RR* )
9	8	3anim3i 1214	. . . 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 10206	. . 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 1005   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8091   RR*cxr 8272   < clt 8273   (,)cioo 10184

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-ioo 10188

This theorem is referenced by:  ivthreinc  15456