Theorem ioo0 10694

Description: An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Assertion

Ref	Expression
ioo0	|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) )

Proof of Theorem ioo0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooval 10310	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2	1	eqeq1d 2247	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> { x e. RR* | ( A < x /\ x < B ) } = (/) ) )
3		xrlttr 10197	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
4	3	3com23 1240	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
5	4	3expa 1234	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
6	5	rexlimdva 2668	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x < B ) -> A < B ) )
7		qbtwnxr 10692	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 10025	. . . . . . . . . 10 |- ( x e. QQ -> x e. RR )
9	8	rexrd 8375	. . . . . . . . 9 |- ( x e. QQ -> x e. RR* )
10	9	anim1i 340	. . . . . . . 8 |- ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A < x /\ x < B ) ) )
11	10	reximi2 2646	. . . . . . 7 |- ( E. x e. QQ ( A < x /\ x < B ) -> E. x e. RR* ( A < x /\ x < B ) )
12	7, 11	syl 14	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. RR* ( A < x /\ x < B ) )
13	12	3expia 1236	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A < x /\ x < B ) ) )
14	6, 13	impbid 129	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x < B ) <-> A < B ) )
15	14	notbid 677	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A < x /\ x < B ) <-> -. A < B ) )
16		rabeq0 3552	. . . . 5 |- ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> A. x e. RR* -. ( A < x /\ x < B ) )
17		ralnex 2538	. . . . 5 |- ( A. x e. RR* -. ( A < x /\ x < B ) <-> -. E. x e. RR* ( A < x /\ x < B ) )
18	16, 17	bitri 184	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A < x /\ x < B ) )
19	18	a1i 9	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A < x /\ x < B ) ) )
20		xrlenlt 8390	. . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
21	20	ancoms 268	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
22	15, 19, 21	3bitr4d 220	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> B <_ A ) )
23	2, 22	bitrd 188	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   (/)c0 3520   class class class wbr 4130  (class class class)co 6085   RR*cxr 8359   < clt 8360   <_ cle 8361   QQcq 10019   (,)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-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

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-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  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-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-ioo 10294

This theorem is used by: (None)