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Theorem elicoelioo 27741
Description: Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Assertion
Ref Expression
elicoelioo  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  <->  ( C  =  A  \/  C  e.  ( A (,) B
) ) ) )

Proof of Theorem elicoelioo
StepHypRef Expression
1 simpl1 999 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  e.  RR* )
2 simpl2 1000 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  B  e.  RR* )
3 simprl 756 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4 elico1 11597 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
54biimpa 484 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
) )
65simp1d 1008 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1227 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  RR* )
85simp2d 1009 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  A  <_  C
)
91, 2, 3, 8syl21anc 1227 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  <_  C
)
101, 2jca 532 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  ( A  e. 
RR*  /\  B  e.  RR* ) )
11 simprr 757 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  C  e.  ( A (,) B ) )
125simp3d 1010 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  <  B
)
1310, 3, 12syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  <  B
)
14 elioo1 11594 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
1514notbid 294 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  C  e.  ( A (,) B )  <->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) ) )
1615biimpa 484 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) )
17 3anan32 985 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
1817notbii 296 . . . . . . . . . . 11  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  -.  (
( C  e.  RR*  /\  C  <  B )  /\  A  <  C
) )
19 imnan 422 . . . . . . . . . . 11  |-  ( ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C )  <->  -.  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
2018, 19bitr4i 252 . . . . . . . . . 10  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2116, 20sylib 196 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2221imp 429 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B
) )  /\  ( C  e.  RR*  /\  C  <  B ) )  ->  -.  A  <  C )
2310, 11, 7, 13, 22syl22anc 1229 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  A  <  C )
24 xeqlelt 27739 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  =  C  <->  ( A  <_  C  /\  -.  A  <  C ) ) )
2524biimpar 485 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  C  /\  -.  A  <  C
) )  ->  A  =  C )
261, 7, 9, 23, 25syl22anc 1229 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  =  C )
2726ex 434 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  A  =  C ) )
28 eqcom 2466 . . . . 5  |-  ( A  =  C  <->  C  =  A )
2927, 28syl6ib 226 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A ) )
30 pm5.6 912 . . . 4  |-  ( ( ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A )  <->  ( C  e.  ( A [,) B
)  ->  ( C  e.  ( A (,) B
)  \/  C  =  A ) ) )
3129, 30sylib 196 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  e.  ( A (,) B )  \/  C  =  A ) ) )
32 orcom 387 . . 3  |-  ( ( C  e.  ( A (,) B )  \/  C  =  A )  <-> 
( C  =  A  \/  C  e.  ( A (,) B ) ) )
3331, 32syl6ib 226 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  =  A  \/  C  e.  ( A (,) B ) ) ) )
34 simpr 461 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  =  A )
35 simpl1 999 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  e.  RR* )
3634, 35eqeltrd 2545 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  RR* )
37 xrleid 11381 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
3835, 37syl 16 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  A )
3938, 34breqtrrd 4482 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  C )
40 simpl3 1001 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <  B )
4134, 40eqbrtrd 4476 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  <  B )
42 simpl2 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  B  e.  RR* )
4335, 42, 4syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
4436, 39, 41, 43mpbir3and 1179 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  ( A [,) B
) )
45 ioossico 11638 . . . . 5  |-  ( A (,) B )  C_  ( A [,) B )
46 simpr 461 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A (,) B
) )
4745, 46sseldi 3497 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A [,) B
) )
4844, 47jaodan 785 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  =  A  \/  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4948ex 434 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  =  A  \/  C  e.  ( A (,) B ) )  ->  C  e.  ( A [,) B ) ) )
5033, 49impbid 191 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  <->  ( C  =  A  \/  C  e.  ( A (,) B
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   RR*cxr 9644    < clt 9645    <_ cle 9646   (,)cioo 11554   [,)cico 11556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-ioo 11558  df-ico 11560
This theorem is referenced by:  xrge0mulc1cn  28076
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