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Theorem ico0 12221
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.
StepHypRef Expression
1 icoval 12213 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = { x e. RR* | ( A <_ x /\ x < B ) } )
21eqeq1d 2624 . 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> { x e. RR* | ( A <_ x /\ x < B ) } = (/) ) )
3 df-ne 2795 . . . . . 6 |- ( { x e. RR* | ( A <_ x /\ x < B ) } =/= (/) <-> -. { x e. RR* | ( A <_ x /\ x < B ) } = (/) )
4 rabn0 3958 . . . . . 6 |- ( { x e. RR* | ( A <_ x /\ x < B ) } =/= (/) <-> E. x e. RR* ( A <_ x /\ x < B ) )
53, 4bitr3i 266 . . . . 5 |- ( -. { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> E. x e. RR* ( A <_ x /\ x < B ) )
6 xrlelttr 11987 . . . . . . . . 9 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
763com23 1271 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
873expa 1265 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
98rexlimdva 3031 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) -> A < B ) )
10 qbtwnxr 12031 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
11 qre 11793 . . . . . . . . . . . . 13 |- ( x e. QQ -> x e. RR )
1211rexrd 10089 . . . . . . . . . . . 12 |- ( x e. QQ -> x e. RR* )
1312a1i 11 . . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( x e. QQ -> x e. RR* ) )
14 simpr1 1067 . . . . . . . . . . . . . . . 16 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> A e. RR* )
15 simpl 473 . . . . . . . . . . . . . . . 16 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> x e. RR* )
16 xrltle 11982 . . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ x e. RR* ) -> ( A < x -> A <_ x ) )
1714, 15, 16syl2anc 693 . . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( A < x -> A <_ x ) )
1817anim1d 588 . . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( A < x /\ x < B ) -> ( A <_ x /\ x < B ) ) )
1913, 18anim12d 586 . . . . . . . . . . . . 13 |- ( ( 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 ) ) ) )
2019ex 450 . . . . . . . . . . . 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 ) ) ) ) )
2112, 20syl 17 . . . . . . . . . . 11 |- ( 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 ) ) ) ) )
2221adantr 481 . . . . . . . . . 10 |- ( ( 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 ) ) ) ) )
2322pm2.43b 55 . . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) )
2423reximdv2 3014 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( E. x e. QQ ( A < x /\ x < B ) -> E. x e. RR* ( A <_ x /\ x < B ) ) )
2510, 24mpd 15 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. RR* ( A <_ x /\ x < B ) )
26253expia 1267 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A <_ x /\ x < B ) ) )
279, 26impbid 202 . . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) <-> A < B ) )
285, 27syl5bb 272 . . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> A < B ) )
29 xrltnle 10105 . . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -. B <_ A ) )
3028, 29bitrd 268 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> -. B <_ A ) )
3130con4bid 307 . 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> B <_ A ) )
322, 31bitrd 268 1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   QQcq 11788   [,)cico 12177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ico 12181
This theorem is referenced by:  icombl  23332  ioombl  23333  difioo  29544  volico  40200  voliooico  40209  voliccico  40216  ovn0lem  40779  ovnhoilem1  40815  hspmbllem1  40840
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