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Theorem gtnelioc 37536
Description: A real number larger than the upper bound of a left open right closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
gtnelioc.a  |-  ( ph  ->  A  e.  RR* )
gtnelioc.b  |-  ( ph  ->  B  e.  RR )
gtnelioc.c  |-  ( ph  ->  C  e.  RR* )
gtnelioc.bltc  |-  ( ph  ->  B  <  C )
Assertion
Ref Expression
gtnelioc  |-  ( ph  ->  -.  C  e.  ( A (,] B ) )

Proof of Theorem gtnelioc
StepHypRef Expression
1 gtnelioc.bltc . . . 4  |-  ( ph  ->  B  <  C )
2 gtnelioc.b . . . . . 6  |-  ( ph  ->  B  e.  RR )
32rexrd 9697 . . . . 5  |-  ( ph  ->  B  e.  RR* )
4 gtnelioc.c . . . . 5  |-  ( ph  ->  C  e.  RR* )
5 xrltnle 9708 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <  C  <->  -.  C  <_  B ) )
63, 4, 5syl2anc 665 . . . 4  |-  ( ph  ->  ( B  <  C  <->  -.  C  <_  B )
)
71, 6mpbid 213 . . 3  |-  ( ph  ->  -.  C  <_  B
)
87intn3an3d 1376 . 2  |-  ( ph  ->  -.  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) )
9 gtnelioc.a . . 3  |-  ( ph  ->  A  e.  RR* )
10 elioc2 11704 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <_  B ) ) )
119, 2, 10syl2anc 665 . 2  |-  ( ph  ->  ( C  e.  ( A (,] B )  <-> 
( C  e.  RR  /\  A  <  C  /\  C  <_  B ) ) )
128, 11mtbird 302 1  |-  ( ph  ->  -.  C  e.  ( A (,] B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ w3a 982    e. wcel 1872   class class class wbr 4423  (class class class)co 6305   RRcr 9545   RR*cxr 9681    < clt 9682    <_ cle 9683   (,]cioc 11643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-pre-lttri 9620  ax-pre-lttrn 9621
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-ioc 11647
This theorem is referenced by:  fourierswlem  38034  fouriersw  38035  etransclem18  38057  etransclem46  38085
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