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Theorem ivthinclemdisj 13960
Description: Lemma for ivthinc 13963. 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 5512 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
21eleq1d 2246 . . . . . . 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 2550 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
54adantr 276 . . . . . . 7  |-  ( (
ph  /\  z  e.  L )  ->  A. x  e.  ( A [,] B
) ( F `  x )  e.  RR )
6 fveq2 5512 . . . . . . . . . . . 12  |-  ( w  =  z  ->  ( F `  w )  =  ( F `  z ) )
76breq1d 4011 . . . . . . . . . . 11  |-  ( w  =  z  ->  (
( F `  w
)  <  U  <->  ( F `  z )  <  U
) )
8 ivthinclem.l . . . . . . . . . . 11  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
97, 8elrab2 2896 . . . . . . . . . 10  |-  ( z  e.  L  <->  ( z  e.  ( A [,] B
)  /\  ( F `  z )  <  U
) )
109biimpi 120 . . . . . . . . 9  |-  ( z  e.  L  ->  (
z  e.  ( A [,] B )  /\  ( F `  z )  <  U ) )
1110adantl 277 . . . . . . . 8  |-  ( (
ph  /\  z  e.  L )  ->  (
z  e.  ( A [,] B )  /\  ( F `  z )  <  U ) )
1211simpld 112 . . . . . . 7  |-  ( (
ph  /\  z  e.  L )  ->  z  e.  ( A [,] B
) )
132, 5, 12rspcdva 2846 . . . . . 6  |-  ( (
ph  /\  z  e.  L )  ->  ( F `  z )  e.  RR )
14 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
1514adantr 276 . . . . . 6  |-  ( (
ph  /\  z  e.  L )  ->  U  e.  RR )
1611simprd 114 . . . . . 6  |-  ( (
ph  /\  z  e.  L )  ->  ( F `  z )  <  U )
1713, 15, 16ltnsymd 8071 . . . . 5  |-  ( (
ph  /\  z  e.  L )  ->  -.  U  <  ( F `  z ) )
1817intnand 931 . . . 4  |-  ( (
ph  /\  z  e.  L )  ->  -.  ( z  e.  ( A [,] B )  /\  U  <  ( F `  z )
) )
196breq2d 4013 . . . . 5  |-  ( w  =  z  ->  ( U  <  ( F `  w )  <->  U  <  ( F `  z ) ) )
20 ivthinclem.r . . . . 5  |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) }
2119, 20elrab2 2896 . . . 4  |-  ( z  e.  R  <->  ( z  e.  ( A [,] B
)  /\  U  <  ( F `  z ) ) )
2218, 21sylnibr 677 . . 3  |-  ( (
ph  /\  z  e.  L )  ->  -.  z  e.  R )
2322ralrimiva 2550 . 2  |-  ( ph  ->  A. z  e.  L  -.  z  e.  R
)
24 disj 3471 . 2  |-  ( ( L  i^i  R )  =  (/)  <->  A. z  e.  L  -.  z  e.  R
)
2523, 24sylibr 134 1  |-  ( ph  ->  ( L  i^i  R
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459    i^i cin 3128    C_ wss 3129   (/)c0 3422   class class class wbr 4001   ` cfv 5213  (class class class)co 5870   CCcc 7804   RRcr 7805    < clt 7986   [,]cicc 9885   -cn->ccncf 13899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-cnex 7897  ax-resscn 7898  ax-pre-ltirr 7918  ax-pre-lttrn 7920
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-xp 4630  df-cnv 4632  df-iota 5175  df-fv 5221  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992
This theorem is referenced by:  ivthinclemex  13962
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