Theorem ioc0 9670

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 9334	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,] B ) = { x e. RR* | ( A < x /\ x <_ B ) } )
2	1	eqeq1d 2096	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> { x e. RR* | ( A < x /\ x <_ B ) } = (/) ) )
3		xrltletr 9270	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A < x /\ x <_ B ) -> A < B ) )
4	3	3com23 1149	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A < x /\ x <_ B ) -> A < B ) )
5	4	3expa 1143	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A < x /\ x <_ B ) -> A < B ) )
6	5	rexlimdva 2489	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x <_ B ) -> A < B ) )
7		qbtwnxr 9665	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 9108	. . . . . . . . . . . 12 |- ( x e. QQ -> x e. RR )
9	8	rexrd 7535	. . . . . . . . . . 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 9266	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ B e. RR* ) -> ( x < B -> x <_ B ) )
12	11	3ad2antr2 1109	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( x < B -> x <_ B ) )
13	12	anim2d 330	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( A < x /\ x < B ) -> ( A < x /\ x <_ B ) ) )
14	10, 13	anim12d 328	. . . . . . . . . . . 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 113	. . . . . . . . . . 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 270	. . . . . . . . 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 51	. . . . . . . 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 2472	. . . . . . 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 1145	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A < x /\ x <_ B ) ) )
22	6, 21	impbid 127	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x <_ B ) <-> A < B ) )
23	22	notbid 627	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A < x /\ x <_ B ) <-> -. A < B ) )
24		rabeq0 3312	. . . . 5 |- ( { x e. RR* | ( A < x /\ x <_ B ) } = (/) <-> A. x e. RR* -. ( A < x /\ x <_ B ) )
25		ralnex 2369	. . . . 5 |- ( A. x e. RR* -. ( A < x /\ x <_ B ) <-> -. E. x e. RR* ( A < x /\ x <_ B ) )
26	24, 25	bitri 182	. . . 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 7549	. . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
29	28	ancoms 264	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
30	23, 27, 29	3bitr4d 218	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x <_ B ) } = (/) <-> B <_ A ) )
31	2, 30	bitrd 186	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 924   = wceq 1289   e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   (/)c0 3286   class class class wbr 3845  (class class class)co 5652   RR*cxr 7519   < clt 7520   <_ cle 7521   QQcq 9102   (,]cioc 9305

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-ioc 9309

This theorem is referenced by: (None)