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Theorem ivthinclemdisj 12776
Description: Lemma for ivthinc 12779. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
Hypotheses
Ref Expression
ivth.1  |-  ( ph  ->  A  e.  RR )
ivth.2  |-  ( ph  ->  B  e.  RR )
ivth.3  |-  ( ph  ->  U  e.  RR )
ivth.4  |-  ( ph  ->  A  <  B )
ivth.5  |-  ( ph  ->  ( A [,] B
)  C_  D )
ivth.7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
ivth.8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
ivth.9  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
ivthinc.i  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  <  ( F `  y )
)
ivthinclem.l  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
ivthinclem.r  |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) }
Assertion
Ref Expression
ivthinclemdisj  |-  ( ph  ->  ( L  i^i  R
)  =  (/) )
Distinct variable groups:    w, A    x, A    w, B    x, B    w, F    x, F    w, U    ph, x
Allowed substitution hints:    ph( y, w)    A( y)    B( y)    D( x, y, w)    R( x, y, w)    U( x, y)    F( y)    L( x, y, w)

Proof of Theorem ivthinclemdisj
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fveq2 5414 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
21eleq1d 2206 . . . . . . 7  |-  ( x  =  z  ->  (
( F `  x
)  e.  RR  <->  ( F `  z )  e.  RR ) )
3 ivth.8 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
43ralrimiva 2503 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
54adantr 274 . . . . . . 7  |-  ( (
ph  /\  z  e.  L )  ->  A. x  e.  ( A [,] B
) ( F `  x )  e.  RR )
6 fveq2 5414 . . . . . . . . . . . 12  |-  ( w  =  z  ->  ( F `  w )  =  ( F `  z ) )
76breq1d 3934 . . . . . . . . . . 11  |-  ( w  =  z  ->  (
( F `  w
)  <  U  <->  ( F `  z )  <  U
) )
8 ivthinclem.l . . . . . . . . . . 11  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
97, 8elrab2 2838 . . . . . . . . . 10  |-  ( z  e.  L  <->  ( z  e.  ( A [,] B
)  /\  ( F `  z )  <  U
) )
109biimpi 119 . . . . . . . . 9  |-  ( z  e.  L  ->  (
z  e.  ( A [,] B )  /\  ( F `  z )  <  U ) )
1110adantl 275 . . . . . . . 8  |-  ( (
ph  /\  z  e.  L )  ->  (
z  e.  ( A [,] B )  /\  ( F `  z )  <  U ) )
1211simpld 111 . . . . . . 7  |-  ( (
ph  /\  z  e.  L )  ->  z  e.  ( A [,] B
) )
132, 5, 12rspcdva 2789 . . . . . 6  |-  ( (
ph  /\  z  e.  L )  ->  ( F `  z )  e.  RR )
14 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
1514adantr 274 . . . . . 6  |-  ( (
ph  /\  z  e.  L )  ->  U  e.  RR )
1611simprd 113 . . . . . 6  |-  ( (
ph  /\  z  e.  L )  ->  ( F `  z )  <  U )
1713, 15, 16ltnsymd 7875 . . . . 5  |-  ( (
ph  /\  z  e.  L )  ->  -.  U  <  ( F `  z ) )
1817intnand 916 . . . 4  |-  ( (
ph  /\  z  e.  L )  ->  -.  ( z  e.  ( A [,] B )  /\  U  <  ( F `  z )
) )
196breq2d 3936 . . . . 5  |-  ( w  =  z  ->  ( U  <  ( F `  w )  <->  U  <  ( F `  z ) ) )
20 ivthinclem.r . . . . 5  |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) }
2119, 20elrab2 2838 . . . 4  |-  ( z  e.  R  <->  ( z  e.  ( A [,] B
)  /\  U  <  ( F `  z ) ) )
2218, 21sylnibr 666 . . 3  |-  ( (
ph  /\  z  e.  L )  ->  -.  z  e.  R )
2322ralrimiva 2503 . 2  |-  ( ph  ->  A. z  e.  L  -.  z  e.  R
)
24 disj 3406 . 2  |-  ( ( L  i^i  R )  =  (/)  <->  A. z  e.  L  -.  z  e.  R
)
2523, 24sylibr 133 1  |-  ( ph  ->  ( L  i^i  R
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2414   {crab 2418    i^i cin 3065    C_ wss 3066   (/)c0 3358   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612    < clt 7793   [,]cicc 9667   -cn->ccncf 12715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725  ax-pre-lttrn 7727
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-iota 5083  df-fv 5126  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799
This theorem is referenced by:  ivthinclemex  12778
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