Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elicc3 Structured version   Unicode version

Theorem elicc3 30758
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 11680 . 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 11455 . . . . . . 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 30753 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  A  <_  B ) )
8 df-ne 2627 . . . . . . . . . 10  |-  ( C  =/=  A  <->  -.  C  =  A )
9 xrleltne 11444 . . . . . . . . . . 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 2438 . . . . . . . . . . . . . 14  |-  ( C  =  B  <->  B  =  C )
1514necon3bbii 2692 . . . . . . . . . . . . 13  |-  ( -.  C  =  B  <->  B  =/=  C )
16 xrleltne 11444 . . . . . . . . . . . . . 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 11449 . . . . . . . . 9  |-  ( A  e.  RR*  ->  A  <_  A )
3837ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  A  <_  A )
39 breq2 4430 . . . . . . . 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 11448 . . . . . . . . . 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 4430 . . . . . . . 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 4429 . . . . . . . 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 11448 . . . . . . . . . . 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 11449 . . . . . . . . 9  |-  ( B  e.  RR*  ->  B  <_  B )
5958ad3antlr 735 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  B  <_  B )
60 breq1 4429 . . . . . . . 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 1870    =/= wne 2625   class class class wbr 4426  (class class class)co 6305   RR*cxr 9673    < clt 9674    <_ cle 9675   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-icc 11642
This theorem is referenced by:  ivthALT  30776
  Copyright terms: Public domain W3C validator