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Theorem elicc3 30922
Description: An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
Assertion
Ref Expression
elicc3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) ) )

Proof of Theorem elicc3
StepHypRef Expression
1 elicc1 11631 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
2 simp1 1005 . . . . 5  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  C  e.  RR* )
32a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  C  e.  RR* ) )
4 xrletr 11406 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  B )  ->  A  <_  B
) )
54exp5o 1224 . . . . . 6  |-  ( A  e.  RR*  ->  ( C  e.  RR*  ->  ( B  e.  RR*  ->  ( A  <_  C  ->  ( C  <_  B  ->  A  <_  B ) ) ) ) )
65com23 81 . . . . 5  |-  ( A  e.  RR*  ->  ( B  e.  RR*  ->  ( C  e.  RR*  ->  ( A  <_  C  ->  ( C  <_  B  ->  A  <_  B ) ) ) ) )
76imp5q 30917 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  A  <_  B ) )
8 df-ne 2601 . . . . . . . . . 10  |-  ( C  =/=  A  <->  -.  C  =  A )
9 xrleltne 11395 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( A  <  C  <->  C  =/=  A ) )
109biimprd 226 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( C  =/=  A  ->  A  <  C ) )
118, 10syl5bir 221 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( -.  C  =  A  ->  A  <  C ) )
12113adant3r3 1216 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  -> 
( -.  C  =  A  ->  A  <  C ) )
1312adantlr 719 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( -.  C  =  A  ->  A  < 
C ) )
14 eqcom 2435 . . . . . . . . . . . . . 14  |-  ( C  =  B  <->  B  =  C )
1514necon3bbii 2648 . . . . . . . . . . . . 13  |-  ( -.  C  =  B  <->  B  =/=  C )
16 xrleltne 11395 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( C  <  B  <->  B  =/=  C ) )
1716biimprd 226 . . . . . . . . . . . . 13  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( B  =/=  C  ->  C  <  B ) )
1815, 17syl5bi 220 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( -.  C  =  B  ->  C  <  B ) )
19183exp 1204 . . . . . . . . . . 11  |-  ( C  e.  RR*  ->  ( B  e.  RR*  ->  ( C  <_  B  ->  ( -.  C  =  B  ->  C  <  B ) ) ) )
2019com12 32 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( C  e.  RR*  ->  ( C  <_  B  ->  ( -.  C  =  B  ->  C  <  B ) ) ) )
2120imp32 434 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  ( C  e.  RR*  /\  C  <_  B ) )  -> 
( -.  C  =  B  ->  C  <  B ) )
22213adantr2 1165 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  -> 
( -.  C  =  B  ->  C  <  B ) )
2322adantll 718 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( -.  C  =  B  ->  C  < 
B ) )
2413, 23anim12d 565 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( ( -.  C  =  A  /\  -.  C  =  B
)  ->  ( A  <  C  /\  C  < 
B ) ) )
2524ex 435 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  -> 
( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) ) ) )
26 df-or 371 . . . . . 6  |-  ( ( C  =  A  \/  ( ( A  < 
C  /\  C  <  B )  \/  C  =  B ) )  <->  ( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
27 3orass 985 . . . . . 6  |-  ( ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B )  <-> 
( C  =  A  \/  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
28 pm5.6 920 . . . . . . 7  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( -.  C  =  A  ->  ( C  =  B  \/  ( A  <  C  /\  C  <  B ) ) ) )
29 orcom 388 . . . . . . . 8  |-  ( ( C  =  B  \/  ( A  <  C  /\  C  <  B ) )  <-> 
( ( A  < 
C  /\  C  <  B )  \/  C  =  B ) )
3029imbi2i 313 . . . . . . 7  |-  ( ( -.  C  =  A  ->  ( C  =  B  \/  ( A  <  C  /\  C  <  B ) ) )  <-> 
( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
3128, 30bitri 252 . . . . . 6  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
3226, 27, 313bitr4ri 281 . . . . 5  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )
3325, 32syl6ib 229 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  -> 
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B ) ) )
343, 7, 333jcad 1186 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  -> 
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) ) )
35 simp1 1005 . . . . 5  |-  ( ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  C  e.  RR* )
3635a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  C  e.  RR* ) )
37 xrleid 11400 . . . . . . . . 9  |-  ( A  e.  RR*  ->  A  <_  A )
3837ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  A  <_  A )
39 breq2 4370 . . . . . . . 8  |-  ( C  =  A  ->  ( A  <_  C  <->  A  <_  A ) )
4038, 39syl5ibrcom 225 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  A  ->  A  <_  C ) )
41 xrltle 11399 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  <  C  ->  A  <_  C ) )
4241adantr 466 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( A  < 
C  ->  A  <_  C ) )
4342adantllr 723 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( A  <  C  ->  A  <_  C ) )
4443adantrd 469 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( A  <  C  /\  C  <  B )  ->  A  <_  C
) )
45 simpr 462 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  A  <_  B )
46 breq2 4370 . . . . . . . 8  |-  ( C  =  B  ->  ( A  <_  C  <->  A  <_  B ) )
4745, 46syl5ibrcom 225 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  B  ->  A  <_  C ) )
4840, 44, 473jaod 1328 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  A  <_  C ) )
4948exp31 607 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  RR*  ->  ( A  <_  B  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  A  <_  C ) ) ) )
50493impd 1219 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  A  <_  C ) )
51 breq1 4369 . . . . . . . 8  |-  ( C  =  A  ->  ( C  <_  B  <->  A  <_  B ) )
5245, 51syl5ibrcom 225 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  A  ->  C  <_  B ) )
53 xrltle 11399 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  <  B  ->  C  <_  B ) )
5453ancoms 454 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( C  <  B  ->  C  <_  B ) )
5554adantld 468 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
( A  <  C  /\  C  <  B )  ->  C  <_  B
) )
5655adantll 718 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* )  ->  ( ( A  < 
C  /\  C  <  B )  ->  C  <_  B ) )
5756adantr 466 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( A  <  C  /\  C  <  B )  ->  C  <_  B
) )
58 xrleid 11400 . . . . . . . . 9  |-  ( B  e.  RR*  ->  B  <_  B )
5958ad3antlr 735 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  B  <_  B )
60 breq1 4369 . . . . . . . 8  |-  ( C  =  B  ->  ( C  <_  B  <->  B  <_  B ) )
6159, 60syl5ibrcom 225 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  B  ->  C  <_  B ) )
6252, 57, 613jaod 1328 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  C  <_  B ) )
6362exp31 607 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  RR*  ->  ( A  <_  B  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  C  <_  B ) ) ) )
64633impd 1219 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  C  <_  B ) )
6536, 50, 643jcad 1186 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) ) )
6634, 65impbid 193 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  <->  ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) ) )
671, 66bitrd 256 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366  (class class class)co 6249   RR*cxr 9625    < clt 9626    <_ cle 9627   [,]cicc 11589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-pre-lttri 9564  ax-pre-lttrn 9565
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-icc 11593
This theorem is referenced by:  ivthALT  30940
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