Theorem ioo0 10439

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 10065	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2	1	eqeq1d 2216	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> { x e. RR* | ( A < x /\ x < B ) } = (/) ) )
3		xrlttr 9952	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
4	3	3com23 1212	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
5	4	3expa 1206	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
6	5	rexlimdva 2625	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x < B ) -> A < B ) )
7		qbtwnxr 10437	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 9781	. . . . . . . . . 10 |- ( x e. QQ -> x e. RR )
9	8	rexrd 8157	. . . . . . . . 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 2604	. . . . . . 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 1208	. . . . 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 669	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A < x /\ x < B ) <-> -. A < B ) )
16		rabeq0 3498	. . . . 5 |- ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> A. x e. RR* -. ( A < x /\ x < B ) )
17		ralnex 2496	. . . . 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 8172	. . . 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 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   {crab 2490   (/)c0 3468   class class class wbr 4059  (class class class)co 5967   RR*cxr 8141   < clt 8142   <_ cle 8143   QQcq 9775   (,)cioo 10045

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-ioo 10049

This theorem is referenced by: (None)