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Theorem elicoelioo 28193
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 1008 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  e.  RR* )
2 simpl2 1009 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  B  e.  RR* )
3 simprl 762 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4 elico1 11668 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
54biimpa 486 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
) )
65simp1d 1017 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1263 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  RR* )
85simp2d 1018 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  A  <_  C
)
91, 2, 3, 8syl21anc 1263 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  <_  C
)
101, 2jca 534 . . . . . . . 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 764 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  C  e.  ( A (,) B ) )
125simp3d 1019 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  <  B
)
1310, 3, 12syl2anc 665 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  <  B
)
14 elioo1 11665 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
1514notbid 295 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  C  e.  ( A (,) B )  <->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) ) )
1615biimpa 486 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) )
17 3anan32 994 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
1817notbii 297 . . . . . . . . . . 11  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  -.  (
( C  e.  RR*  /\  C  <  B )  /\  A  <  C
) )
19 imnan 423 . . . . . . . . . . 11  |-  ( ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C )  <->  -.  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
2018, 19bitr4i 255 . . . . . . . . . 10  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2116, 20sylib 199 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2221imp 430 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B
) )  /\  ( C  e.  RR*  /\  C  <  B ) )  ->  -.  A  <  C )
2310, 11, 7, 13, 22syl22anc 1265 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  A  <  C )
24 xeqlelt 28191 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  =  C  <->  ( A  <_  C  /\  -.  A  <  C ) ) )
2524biimpar 487 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  C  /\  -.  A  <  C
) )  ->  A  =  C )
261, 7, 9, 23, 25syl22anc 1265 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  =  C )
2726ex 435 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  A  =  C ) )
28 eqcom 2429 . . . . 5  |-  ( A  =  C  <->  C  =  A )
2927, 28syl6ib 229 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A ) )
30 pm5.6 920 . . . 4  |-  ( ( ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A )  <->  ( C  e.  ( A [,) B
)  ->  ( C  e.  ( A (,) B
)  \/  C  =  A ) ) )
3129, 30sylib 199 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  e.  ( A (,) B )  \/  C  =  A ) ) )
32 orcom 388 . . 3  |-  ( ( C  e.  ( A (,) B )  \/  C  =  A )  <-> 
( C  =  A  \/  C  e.  ( A (,) B ) ) )
3331, 32syl6ib 229 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  =  A  \/  C  e.  ( A (,) B ) ) ) )
34 simpr 462 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  =  A )
35 simpl1 1008 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  e.  RR* )
3634, 35eqeltrd 2508 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  RR* )
37 xrleid 11438 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
3835, 37syl 17 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  A )
3938, 34breqtrrd 4443 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  C )
40 simpl3 1010 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <  B )
4134, 40eqbrtrd 4437 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  <  B )
42 simpl2 1009 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  B  e.  RR* )
4335, 42, 4syl2anc 665 . . . . 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 1188 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  ( A [,) B
) )
45 ioossico 11712 . . . . 5  |-  ( A (,) B )  C_  ( A [,) B )
46 simpr 462 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A (,) B
) )
4745, 46sseldi 3459 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A [,) B
) )
4844, 47jaodan 792 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  =  A  \/  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4948ex 435 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  =  A  \/  C  e.  ( A (,) B ) )  ->  C  e.  ( A [,) B ) ) )
5033, 49impbid 193 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   class class class wbr 4417  (class class class)co 6296   RR*cxr 9663    < clt 9664    <_ cle 9665   (,)cioo 11624   [,)cico 11626
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-pre-lttri 9602  ax-pre-lttrn 9603
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 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-ioo 11628  df-ico 11630
This theorem is referenced by:  xrge0mulc1cn  28583
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