Theorem elioo4g 10168

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 10161	. . . . 5 |- ( C e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* ) )
2		elioore 10146	. . . . 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 1006	. . . 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 10162	. . 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 8224	. . . . 5 |- ( C e. RR -> C e. RR* )
9	8	3anim3i 1213	. . . 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 10144	. . 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 1004   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   RR*cxr 8212   < clt 8213   (,)cioo 10122

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-ioo 10126

This theorem is referenced by:  ivthreinc  15368