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Theorem elicc3 26252
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 10944 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
2 simp1 957 . . . . 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 10732 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  B )  ->  A  <_  B
) )
54exp5o 1172 . . . . . 6  |-  ( A  e.  RR*  ->  ( C  e.  RR*  ->  ( B  e.  RR*  ->  ( A  <_  C  ->  ( C  <_  B  ->  A  <_  B ) ) ) ) )
65com23 74 . . . . 5  |-  ( A  e.  RR*  ->  ( B  e.  RR*  ->  ( C  e.  RR*  ->  ( A  <_  C  ->  ( C  <_  B  ->  A  <_  B ) ) ) ) )
76imp5q 26247 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  A  <_  B ) )
8 df-ne 2595 . . . . . . . . . 10  |-  ( C  =/=  A  <->  -.  C  =  A )
9 xrleltne 10722 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( A  <  C  <->  C  =/=  A ) )
109biimprd 215 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( C  =/=  A  ->  A  <  C ) )
118, 10syl5bir 210 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( -.  C  =  A  ->  A  <  C ) )
12113adant3r3 1164 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  -> 
( -.  C  =  A  ->  A  <  C ) )
1312adantlr 696 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( -.  C  =  A  ->  A  < 
C ) )
14 eqcom 2432 . . . . . . . . . . . . . 14  |-  ( C  =  B  <->  B  =  C )
1514necon3bbii 2624 . . . . . . . . . . . . 13  |-  ( -.  C  =  B  <->  B  =/=  C )
16 xrleltne 10722 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( C  <  B  <->  B  =/=  C ) )
1716biimprd 215 . . . . . . . . . . . . 13  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( B  =/=  C  ->  C  <  B ) )
1815, 17syl5bi 209 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( -.  C  =  B  ->  C  <  B ) )
19183exp 1152 . . . . . . . . . . 11  |-  ( C  e.  RR*  ->  ( B  e.  RR*  ->  ( C  <_  B  ->  ( -.  C  =  B  ->  C  <  B ) ) ) )
2019com12 29 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( C  e.  RR*  ->  ( C  <_  B  ->  ( -.  C  =  B  ->  C  <  B ) ) ) )
2120imp32 423 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  ( C  e.  RR*  /\  C  <_  B ) )  -> 
( -.  C  =  B  ->  C  <  B ) )
22213adantr2 1117 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  -> 
( -.  C  =  B  ->  C  <  B ) )
2322adantll 695 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( -.  C  =  B  ->  C  < 
B ) )
2413, 23anim12d 547 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( ( -.  C  =  A  /\  -.  C  =  B
)  ->  ( A  <  C  /\  C  < 
B ) ) )
2524ex 424 . . . . 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 360 . . . . . 6  |-  ( ( C  =  A  \/  ( ( A  < 
C  /\  C  <  B )  \/  C  =  B ) )  <->  ( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
27 3orass 939 . . . . . 6  |-  ( ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B )  <-> 
( C  =  A  \/  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
28 pm5.6 879 . . . . . . 7  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( -.  C  =  A  ->  ( C  =  B  \/  ( A  <  C  /\  C  <  B ) ) ) )
29 orcom 377 . . . . . . . 8  |-  ( ( C  =  B  \/  ( A  <  C  /\  C  <  B ) )  <-> 
( ( A  < 
C  /\  C  <  B )  \/  C  =  B ) )
3029imbi2i 304 . . . . . . 7  |-  ( ( -.  C  =  A  ->  ( C  =  B  \/  ( A  <  C  /\  C  <  B ) ) )  <-> 
( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
3128, 30bitri 241 . . . . . 6  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
3226, 27, 313bitr4ri 270 . . . . 5  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )
3325, 32syl6ib 218 . . . 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 1135 . . 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 957 . . . . 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 10727 . . . . . . . . 9  |-  ( A  e.  RR*  ->  A  <_  A )
3837ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  A  <_  A )
39 breq2 4203 . . . . . . . 8  |-  ( C  =  A  ->  ( A  <_  C  <->  A  <_  A ) )
4038, 39syl5ibrcom 214 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  A  ->  A  <_  C ) )
41 xrltle 10726 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  <  C  ->  A  <_  C ) )
4241adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( A  < 
C  ->  A  <_  C ) )
4342adantllr 700 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( A  <  C  ->  A  <_  C ) )
4443adantrd 455 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( A  <  C  /\  C  <  B )  ->  A  <_  C
) )
45 simpr 448 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  A  <_  B )
46 breq2 4203 . . . . . . . 8  |-  ( C  =  B  ->  ( A  <_  C  <->  A  <_  B ) )
4745, 46syl5ibrcom 214 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  B  ->  A  <_  C ) )
4840, 44, 473jaod 1248 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  A  <_  C ) )
4948exp31 588 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  RR*  ->  ( A  <_  B  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  A  <_  C ) ) ) )
50493impd 1167 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  A  <_  C ) )
51 breq1 4202 . . . . . . . 8  |-  ( C  =  A  ->  ( C  <_  B  <->  A  <_  B ) )
5245, 51syl5ibrcom 214 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  A  ->  C  <_  B ) )
53 xrltle 10726 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  <  B  ->  C  <_  B ) )
5453ancoms 440 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( C  <  B  ->  C  <_  B ) )
5554adantld 454 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
( A  <  C  /\  C  <  B )  ->  C  <_  B
) )
5655adantll 695 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* )  ->  ( ( A  < 
C  /\  C  <  B )  ->  C  <_  B ) )
5756adantr 452 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( A  <  C  /\  C  <  B )  ->  C  <_  B
) )
58 xrleid 10727 . . . . . . . . 9  |-  ( B  e.  RR*  ->  B  <_  B )
5958ad3antlr 712 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  B  <_  B )
60 breq1 4202 . . . . . . . 8  |-  ( C  =  B  ->  ( C  <_  B  <->  B  <_  B ) )
6159, 60syl5ibrcom 214 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  B  ->  C  <_  B ) )
6252, 57, 613jaod 1248 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  C  <_  B ) )
6362exp31 588 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  RR*  ->  ( A  <_  B  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  C  <_  B ) ) ) )
64633impd 1167 . . . 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 1135 . . 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 184 . 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 245 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   class class class wbr 4199  (class class class)co 6067   RR*cxr 9103    < clt 9104    <_ cle 9105   [,]cicc 10903
This theorem is referenced by:  ivthALT  26270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-pre-lttri 9048  ax-pre-lttrn 9049
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-po 4490  df-so 4491  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-icc 10907
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