Theorem elioo3g 6405

Description: Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show A e. RR* and B e. RR*.

Assertion

Ref	Expression
elioo3g	|- (B e. D -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR* /\ C e. RR*) /\ (A < C /\ C < B))))

Proof of Theorem elioo3g

Step	Hyp	Ref	Expression
1		elioo1 6403	. . . . 5 |- ((A e. RR* /\ B e. RR*) -> (C e. (A(,)B) <-> (C e. RR* /\ A < C /\ C < B)))
2		ibar 654	. . . . . 6 |- ((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)))))
3		3anass 791	. . . . . 6 |- ((C e. RR* /\ A < C /\ C < B) <-> (C e. RR* /\ (A < C /\ C < B)))
4	2, 3	syl5bb 543	. . . . 5 |- ((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)))))
5	1, 4	bitrd 539	. . . 4 |- ((A e. RR* /\ B e. RR*) -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ (A < C /\ C < B)))))
6	5	adantl 397	. . 3 |- ((B e. D /\ (A e. RR* /\ B e. RR*)) -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ (A < C /\ C < B)))))
7		xrex 5557	. . . . . . . 8 |- RR* e. V
8	7	rabex 2780	. . . . . . 7 |- {w e. RR* | (x < w /\ w < y)} e. V
9		df-ioo 6386	. . . . . . 7 |- (,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
10	8, 9	dmoprab2 4181	. . . . . 6 |- dom (,) = (RR* X. RR*)
11		ndmoprg 4101	. . . . . 6 |- ((dom (,) = (RR* X. RR*) /\ B e. D /\ -. (A e. RR* /\ B e. RR*)) -> (A(,)B) = (/))
12	10, 11	mp3an1 915	. . . . 5 |- ((B e. D /\ -. (A e. RR* /\ B e. RR*)) -> (A(,)B) = (/))
13	12	eleq2d 1588	. . . 4 |- ((B e. D /\ -. (A e. RR* /\ B e. RR*)) -> (C e. (A(,)B) <-> C e. (/)))
14		noel 2335	. . . . . . 7 |- -. C e. (/)
15	14	pm2.21i 80	. . . . . 6 |- (C e. (/) -> (A e. RR* /\ B e. RR*))
16		pm3.26 326	. . . . . 6 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ (A < C /\ C < B))) -> (A e. RR* /\ B e. RR*))
17	15, 16	pm5.21ni 690	. . . . 5 |- (-. (A e. RR* /\ B e. RR*) -> (C e. (/) <-> ((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ (A < C /\ C < B)))))
18	17	adantl 397	. . . 4 |- ((B e. D /\ -. (A e. RR* /\ B e. RR*)) -> (C e. (/) <-> ((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ (A < C /\ C < B)))))
19	13, 18	bitrd 539	. . 3 |- ((B e. D /\ -. (A e. RR* /\ B e. RR*)) -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ (A < C /\ C < B)))))
20	6, 19	pm2.61dan 488	. 2 |- (B e. D -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ (A < C /\ C < B)))))
21		df-3an 789	. . . 4 |- ((A e. RR* /\ B e. RR* /\ C e. RR*) <-> ((A e. RR* /\ B e. RR*) /\ C e. RR*))
22	21	anbi1i 492	. . 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)))
23		anass 450	. . 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))))
24	22, 23	bitr2i 181	. 2 |- (((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)))
25	20, 24	syl6bb 547	1 |- (B e. D -> (C e. (A(,)B) <-> ((A e. RR* /\ B e. RR* /\ C e. RR*) /\ (A < C /\ C < B))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   /\ wa 230   /\ w3a 787   = wceq 997   e. wcel 999  {crab 1695  (/)c0 2331   class class class wbr 2674   X. cxp 3225  dom cdm 3227  (class class class)co 4021  RR*cxr 5550   < clt 5551  (,)cioo 6382

This theorem is referenced by:  elioo4g 6410  iint 10716

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922  ax-inf2 4687

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-reu 1698  df-rab 1699  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-pss 2106  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-om 3189  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-fv 3255  df-rdg 3990  df-opr 4023  df-oprab 4024  df-1st 4137  df-2nd 4138  df-1o 4191  df-oadd 4193  df-omul 4194  df-er 4319  df-ec 4321  df-qs 4324  df-ni 5065  df-pli 5066  df-mi 5067  df-lti 5068  df-plpq 5100  df-mpq 5101  df-enq 5102  df-nq 5103  df-plq 5104  df-mq 5105  df-rq 5106  df-ltq 5107  df-1q 5108  df-np 5151  df-1p 5152  df-enr 5231  df-nr 5232  df-0r 5236  df-c 5305  df-r 5309  df-xr 5554  df-ioo 6386

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