Theorem ico0 10584

Description: An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Assertion

Ref	Expression
ico0	|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )

Proof of Theorem ico0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icoval 10215	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = { x e. RR* | ( A <_ x /\ x < B ) } )
2	1	eqeq1d 2240	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> { x e. RR* | ( A <_ x /\ x < B ) } = (/) ) )
3		xrlelttr 10102	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
4	3	3com23 1236	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
5	4	3expa 1230	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
6	5	rexlimdva 2651	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) -> A < B ) )
7		qbtwnxr 10580	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 9920	. . . . . . . . . . . 12 |- ( x e. QQ -> x e. RR )
9	8	rexrd 8288	. . . . . . . . . . 11 |- ( x e. QQ -> x e. RR* )
10	9	a1i 9	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( x e. QQ -> x e. RR* ) )
11		simpr1 1030	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> A e. RR* )
12		simpl 109	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> x e. RR* )
13		xrltle 10094	. . . . . . . . . . . . . . 15 |- ( ( A e. RR* /\ x e. RR* ) -> ( A < x -> A <_ x ) )
14	11, 12, 13	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( A < x -> A <_ x ) )
15	14	anim1d 336	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( A < x /\ x < B ) -> ( A <_ x /\ x < B ) ) )
16	10, 15	anim12d 335	. . . . . . . . . . . 12 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) )
17	16	ex 115	. . . . . . . . . . 11 |- ( x e. RR* -> ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) ) )
18	9, 17	syl 14	. . . . . . . . . 10 |- ( x e. QQ -> ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) ) )
19	18	adantr 276	. . . . . . . . 9 |- ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) ) )
20	19	pm2.43b 52	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) )
21	20	reximdv2 2632	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( E. x e. QQ ( A < x /\ x < B ) -> E. x e. RR* ( A <_ x /\ x < B ) ) )
22	7, 21	mpd 13	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. RR* ( A <_ x /\ x < B ) )
23	22	3expia 1232	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A <_ x /\ x < B ) ) )
24	6, 23	impbid 129	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) <-> A < B ) )
25	24	notbid 673	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A <_ x /\ x < B ) <-> -. A < B ) )
26		rabeq0 3526	. . . . 5 |- ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> A. x e. RR* -. ( A <_ x /\ x < B ) )
27		ralnex 2521	. . . . 5 |- ( A. x e. RR* -. ( A <_ x /\ x < B ) <-> -. E. x e. RR* ( A <_ x /\ x < B ) )
28	26, 27	bitri 184	. . . 4 |- ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A <_ x /\ x < B ) )
29	28	a1i 9	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A <_ x /\ x < B ) ) )
30		xrlenlt 8303	. . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
31	30	ancoms 268	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
32	25, 29, 31	3bitr4d 220	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> B <_ A ) )
33	2, 32	bitrd 188	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   {crab 2515   (/)c0 3496   class class class wbr 4093  (class class class)co 6028   RR*cxr 8272   < clt 8273   <_ cle 8274   QQcq 9914   [,)cico 10186

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-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211

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-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  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-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-ico 10190

This theorem is referenced by: (None)