Theorem ioc0 10040

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

Assertion

Ref	Expression
ioc0	|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> B <_ A ) )

Proof of Theorem ioc0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iocval 9701	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,] B ) = { x e. RR* | ( A < x /\ x <_ B ) } )
2	1	eqeq1d 2148	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> { x e. RR* | ( A < x /\ x <_ B ) } = (/) ) )
3		xrltletr 9590	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A < x /\ x <_ B ) -> A < B ) )
4	3	3com23 1187	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A < x /\ x <_ B ) -> A < B ) )
5	4	3expa 1181	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A < x /\ x <_ B ) -> A < B ) )
6	5	rexlimdva 2549	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x <_ B ) -> A < B ) )
7		qbtwnxr 10035	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 9417	. . . . . . . . . . . 12 |- ( x e. QQ -> x e. RR )
9	8	rexrd 7815	. . . . . . . . . . 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		xrltle 9584	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ B e. RR* ) -> ( x < B -> x <_ B ) )
12	11	3ad2antr2 1147	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( x < B -> x <_ B ) )
13	12	anim2d 335	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( A < x /\ x < B ) -> ( A < x /\ x <_ B ) ) )
14	10, 13	anim12d 333	. . . . . . . . . . . 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 ) ) ) )
15	14	ex 114	. . . . . . . . . . 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 ) ) ) ) )
16	9, 15	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 ) ) ) ) )
17	16	adantr 274	. . . . . . . . 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 ) ) ) ) )
18	17	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 ) ) ) )
19	18	reximdv2 2531	. . . . . . 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 ) ) )
20	7, 19	mpd 13	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. RR* ( A < x /\ x <_ B ) )
21	20	3expia 1183	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A < x /\ x <_ B ) ) )
22	6, 21	impbid 128	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x <_ B ) <-> A < B ) )
23	22	notbid 656	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A < x /\ x <_ B ) <-> -. A < B ) )
24		rabeq0 3392	. . . . 5 |- ( { x e. RR* | ( A < x /\ x <_ B ) } = (/) <-> A. x e. RR* -. ( A < x /\ x <_ B ) )
25		ralnex 2426	. . . . 5 |- ( A. x e. RR* -. ( A < x /\ x <_ B ) <-> -. E. x e. RR* ( A < x /\ x <_ B ) )
26	24, 25	bitri 183	. . . 4 |- ( { x e. RR* | ( A < x /\ x <_ B ) } = (/) <-> -. E. x e. RR* ( A < x /\ x <_ B ) )
27	26	a1i 9	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x <_ B ) } = (/) <-> -. E. x e. RR* ( A < x /\ x <_ B ) ) )
28		xrlenlt 7829	. . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
29	28	ancoms 266	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
30	23, 27, 29	3bitr4d 219	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x <_ B ) } = (/) <-> B <_ A ) )
31	2, 30	bitrd 187	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   (/)c0 3363   class class class wbr 3929  (class class class)co 5774   RR*cxr 7799   < clt 7800   <_ cle 7801   QQcq 9411   (,]cioc 9672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ioc 9676

This theorem is referenced by: (None)