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Theorem rpnnen3lem 26786
Description: Lemma for rpnnen3 26787. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Assertion
Ref Expression
rpnnen3lem  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  a  <  b
)  ->  { c  e.  QQ  |  c  < 
a }  =/=  {
c  e.  QQ  | 
c  <  b }
)
Distinct variable group:    a, b, c

Proof of Theorem rpnnen3lem
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 qbtwnre 10710 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  ->  E. d  e.  QQ  ( a  < 
d  /\  d  <  b ) )
2 simp2 958 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  e.  QQ )
3 simp3r 986 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  <  b
)
4 breq1 4149 . . . . . . . 8  |-  ( c  =  d  ->  (
c  <  b  <->  d  <  b ) )
54elrab 3028 . . . . . . 7  |-  ( d  e.  { c  e.  QQ  |  c  < 
b }  <->  ( d  e.  QQ  /\  d  < 
b ) )
62, 3, 5sylanbrc 646 . . . . . 6  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  e.  {
c  e.  QQ  | 
c  <  b }
)
7 simp11 987 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  a  e.  RR )
8 qre 10504 . . . . . . . . . 10  |-  ( d  e.  QQ  ->  d  e.  RR )
983ad2ant2 979 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  e.  RR )
10 simp3l 985 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  a  <  d
)
117, 9, 10ltnsymd 9147 . . . . . . . 8  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  -.  d  <  a )
1211intnand 883 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  -.  ( d  e.  QQ  /\  d  < 
a ) )
13 breq1 4149 . . . . . . . 8  |-  ( c  =  d  ->  (
c  <  a  <->  d  <  a ) )
1413elrab 3028 . . . . . . 7  |-  ( d  e.  { c  e.  QQ  |  c  < 
a }  <->  ( d  e.  QQ  /\  d  < 
a ) )
1512, 14sylnibr 297 . . . . . 6  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  -.  d  e.  { c  e.  QQ  | 
c  <  a }
)
16 nelne1 2632 . . . . . 6  |-  ( ( d  e.  { c  e.  QQ  |  c  <  b }  /\  -.  d  e.  { c  e.  QQ  |  c  <  a } )  ->  { c  e.  QQ  |  c  < 
b }  =/=  {
c  e.  QQ  | 
c  <  a }
)
176, 15, 16syl2anc 643 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  { c  e.  QQ  |  c  < 
b }  =/=  {
c  e.  QQ  | 
c  <  a }
)
1817necomd 2626 . . . 4  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  { c  e.  QQ  |  c  < 
a }  =/=  {
c  e.  QQ  | 
c  <  b }
)
1918rexlimdv3a 2768 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  ->  ( E. d  e.  QQ  ( a  <  d  /\  d  <  b )  ->  { c  e.  QQ  |  c  < 
a }  =/=  {
c  e.  QQ  | 
c  <  b }
) )
201, 19mpd 15 . 2  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  ->  { c  e.  QQ  |  c  <  a }  =/=  { c  e.  QQ  | 
c  <  b }
)
21203expa 1153 1  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  a  <  b
)  ->  { c  e.  QQ  |  c  < 
a }  =/=  {
c  e.  QQ  | 
c  <  b }
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1717    =/= wne 2543   E.wrex 2643   {crab 2646   class class class wbr 4146   RRcr 8915    < clt 9046   QQcq 10499
This theorem is referenced by:  rpnnen3  26787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500
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