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Theorem elicc3 26228
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 10700 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
2 simp1 955 . . . . 5  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  C  e.  RR* )
32a1i 10 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  C  e.  RR* ) )
4 xrletr 10489 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <_  C  /\  C  <_  B )  ->  A  <_  B
) )
54exp5o 1170 . . . . . 6  |-  ( A  e.  RR*  ->  ( C  e.  RR*  ->  ( B  e.  RR*  ->  ( A  <_  C  ->  ( C  <_  B  ->  A  <_  B ) ) ) ) )
65com23 72 . . . . 5  |-  ( A  e.  RR*  ->  ( B  e.  RR*  ->  ( C  e.  RR*  ->  ( A  <_  C  ->  ( C  <_  B  ->  A  <_  B ) ) ) ) )
76imp5q 26221 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  A  <_  B ) )
8 df-ne 2448 . . . . . . . . . 10  |-  ( C  =/=  A  <->  -.  C  =  A )
9 xrleltne 10479 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( A  <  C  <->  C  =/=  A ) )
109biimprd 214 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( C  =/=  A  ->  A  <  C ) )
118, 10syl5bir 209 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  A  <_  C )  ->  ( -.  C  =  A  ->  A  <  C ) )
12113adant3r3 1162 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  -> 
( -.  C  =  A  ->  A  <  C ) )
1312adantlr 695 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( -.  C  =  A  ->  A  < 
C ) )
14 eqcom 2285 . . . . . . . . . . . . . 14  |-  ( C  =  B  <->  B  =  C )
1514necon3bbii 2477 . . . . . . . . . . . . 13  |-  ( -.  C  =  B  <->  B  =/=  C )
16 xrleltne 10479 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( C  <  B  <->  B  =/=  C ) )
1716biimprd 214 . . . . . . . . . . . . 13  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( B  =/=  C  ->  C  <  B ) )
1815, 17syl5bi 208 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  C  <_  B )  ->  ( -.  C  =  B  ->  C  <  B ) )
19183exp 1150 . . . . . . . . . . 11  |-  ( C  e.  RR*  ->  ( B  e.  RR*  ->  ( C  <_  B  ->  ( -.  C  =  B  ->  C  <  B ) ) ) )
2019com12 27 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( C  e.  RR*  ->  ( C  <_  B  ->  ( -.  C  =  B  ->  C  <  B ) ) ) )
2120imp32 422 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  ( C  e.  RR*  /\  C  <_  B ) )  -> 
( -.  C  =  B  ->  C  <  B ) )
22213adantr2 1115 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  -> 
( -.  C  =  B  ->  C  <  B ) )
2322adantll 694 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( -.  C  =  B  ->  C  < 
B ) )
2413, 23anim12d 546 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) )  ->  ( ( -.  C  =  A  /\  -.  C  =  B
)  ->  ( A  <  C  /\  C  < 
B ) ) )
2524ex 423 . . . . 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 359 . . . . . 6  |-  ( ( C  =  A  \/  ( ( A  < 
C  /\  C  <  B )  \/  C  =  B ) )  <->  ( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
27 3orass 937 . . . . . 6  |-  ( ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B )  <-> 
( C  =  A  \/  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
28 pm5.6 878 . . . . . . 7  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( -.  C  =  A  ->  ( C  =  B  \/  ( A  <  C  /\  C  <  B ) ) ) )
29 orcom 376 . . . . . . . 8  |-  ( ( C  =  B  \/  ( A  <  C  /\  C  <  B ) )  <-> 
( ( A  < 
C  /\  C  <  B )  \/  C  =  B ) )
3029imbi2i 303 . . . . . . 7  |-  ( ( -.  C  =  A  ->  ( C  =  B  \/  ( A  <  C  /\  C  <  B ) ) )  <-> 
( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
3128, 30bitri 240 . . . . . 6  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( -.  C  =  A  ->  ( ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) )
3226, 27, 313bitr4ri 269 . . . . 5  |-  ( ( ( -.  C  =  A  /\  -.  C  =  B )  ->  ( A  <  C  /\  C  <  B ) )  <->  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )
3325, 32syl6ib 217 . . . 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 1133 . . 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 955 . . . . 5  |-  ( ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  C  e.  RR* )
3635a1i 10 . . . 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 10484 . . . . . . . . 9  |-  ( A  e.  RR*  ->  A  <_  A )
3837ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  A  <_  A )
39 breq2 4027 . . . . . . . 8  |-  ( C  =  A  ->  ( A  <_  C  <->  A  <_  A ) )
4038, 39syl5ibrcom 213 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  A  ->  A  <_  C ) )
41 xrltle 10483 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  <  C  ->  A  <_  C ) )
4241adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( A  < 
C  ->  A  <_  C ) )
4342adantllr 699 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( A  <  C  ->  A  <_  C ) )
4443adantrd 454 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( A  <  C  /\  C  <  B )  ->  A  <_  C
) )
45 simpr 447 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  A  <_  B )
46 breq2 4027 . . . . . . . 8  |-  ( C  =  B  ->  ( A  <_  C  <->  A  <_  B ) )
4745, 46syl5ibrcom 213 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  B  ->  A  <_  C ) )
4840, 44, 473jaod 1246 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  A  <_  C ) )
4948exp31 587 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  RR*  ->  ( A  <_  B  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  A  <_  C ) ) ) )
50493impd 1165 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  A  <_  C ) )
51 breq1 4026 . . . . . . . 8  |-  ( C  =  A  ->  ( C  <_  B  <->  A  <_  B ) )
5245, 51syl5ibrcom 213 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  A  ->  C  <_  B ) )
53 xrltle 10483 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  <  B  ->  C  <_  B ) )
5453ancoms 439 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( C  <  B  ->  C  <_  B ) )
5554adantld 453 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
( A  <  C  /\  C  <  B )  ->  C  <_  B
) )
5655adantll 694 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* )  ->  ( ( A  < 
C  /\  C  <  B )  ->  C  <_  B ) )
5756adantr 451 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( A  <  C  /\  C  <  B )  ->  C  <_  B
) )
58 xrleid 10484 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  B  <_  B )
5958ad2antlr 707 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* )  ->  B  <_  B )
6059adantr 451 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  B  <_  B )
61 breq1 4026 . . . . . . . 8  |-  ( C  =  B  ->  ( C  <_  B  <->  B  <_  B ) )
6260, 61syl5ibrcom 213 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  ( C  =  B  ->  C  <_  B ) )
6352, 57, 623jaod 1246 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  C  e. 
RR* )  /\  A  <_  B )  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  C  <_  B ) )
6463exp31 587 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  RR*  ->  ( A  <_  B  ->  (
( C  =  A  \/  ( A  < 
C  /\  C  <  B )  \/  C  =  B )  ->  C  <_  B ) ) ) )
65643impd 1165 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) )  ->  C  <_  B ) )
6636, 50, 653jcad 1133 . . 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 ) ) )
6734, 66impbid 183 . 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 ) ) ) )
681, 67bitrd 244 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   RR*cxr 8866    < clt 8867    <_ cle 8868   [,]cicc 10659
This theorem is referenced by:  ivthALT  26258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
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