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Theorem icodisj 8630
Description: End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
icodisj  RR*  RR*  C 
RR*  [,)  i^i  [,) C  (/)

Proof of Theorem icodisj
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3120 . . . 4  [,)  i^i  [,) C  [,)  [,) C
2 elico1 8562 . . . . . . . . . 10  RR*  RR*  [,)  RR*  <_  <
323adant3 923 . . . . . . . . 9  RR*  RR*  C 
RR*  [,)  RR*  <_  <
43biimpa 280 . . . . . . . 8  RR*  RR*  C  RR*  [,)  RR*  <_  <
54simp3d 917 . . . . . . 7  RR*  RR*  C  RR*  [,)  <
65adantrr 448 . . . . . 6  RR*  RR*  C  RR*  [,)  [,) C  <
7 elico1 8562 . . . . . . . . . . 11  RR*  C  RR*  [,) C  RR*  <_  <  C
873adant1 921 . . . . . . . . . 10  RR*  RR*  C 
RR*  [,) C  RR*  <_  <  C
98biimpa 280 . . . . . . . . 9  RR*  RR*  C  RR*  [,) C  RR*  <_  < 
C
109simp2d 916 . . . . . . . 8  RR*  RR*  C  RR*  [,) C  <_
11 simpl2 907 . . . . . . . . 9  RR*  RR*  C  RR*  [,) C  RR*
129simp1d 915 . . . . . . . . 9  RR*  RR*  C  RR*  [,) C  RR*
13 xrlenlt 6881 . . . . . . . . 9  RR*  RR*  <_  <
1411, 12, 13syl2anc 391 . . . . . . . 8  RR*  RR*  C  RR*  [,) C  <_  <
1510, 14mpbid 135 . . . . . . 7  RR*  RR*  C  RR*  [,) C  <
1615adantrl 447 . . . . . 6  RR*  RR*  C  RR*  [,)  [,) C  <
176, 16pm2.65da 586 . . . . 5  RR*  RR*  C 
RR*  [,)  [,) C
1817pm2.21d 549 . . . 4  RR*  RR*  C 
RR*  [,)  [,) C  (/)
191, 18syl5bi 141 . . 3  RR*  RR*  C 
RR*  [,)  i^i  [,) C  (/)
2019ssrdv 2945 . 2  RR*  RR*  C 
RR*  [,)  i^i  [,) C  C_  (/)
21 ss0 3251 . 2  [,)  i^i  [,) C  C_  (/)  [,)  i^i  [,) C  (/)
2220, 21syl 14 1  RR*  RR*  C 
RR*  [,)  i^i  [,) C  (/)
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390    i^i cin 2910    C_ wss 2911   (/)c0 3218   class class class wbr 3755  (class class class)co 5455   RR*cxr 6856    < clt 6857    <_ cle 6858   [,)cico 8529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-pnf 6859  df-mnf 6860  df-xr 6861  df-le 6863  df-ico 8533
This theorem is referenced by: (None)
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