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Theorem rpnnen3lem 27124
Description: Lemma for rpnnen3 27125. (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 10526 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  ->  E. d  e.  QQ  ( a  < 
d  /\  d  <  b ) )
2 simp2 956 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  e.  QQ )
3 simp3r 984 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  <  b
)
4 breq1 4026 . . . . . . . 8  |-  ( c  =  d  ->  (
c  <  b  <->  d  <  b ) )
54elrab 2923 . . . . . . 7  |-  ( d  e.  { c  e.  QQ  |  c  < 
b }  <->  ( d  e.  QQ  /\  d  < 
b ) )
62, 3, 5sylanbrc 645 . . . . . 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 985 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  a  e.  RR )
8 qre 10321 . . . . . . . . . 10  |-  ( d  e.  QQ  ->  d  e.  RR )
983ad2ant2 977 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  e.  RR )
10 simp3l 983 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  a  <  d
)
117, 9, 10ltnsymd 8968 . . . . . . . 8  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  -.  d  <  a )
1211intnand 882 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  -.  ( d  e.  QQ  /\  d  < 
a ) )
13 breq1 4026 . . . . . . . 8  |-  ( c  =  d  ->  (
c  <  a  <->  d  <  a ) )
1413elrab 2923 . . . . . . 7  |-  ( d  e.  { c  e.  QQ  |  c  < 
a }  <->  ( d  e.  QQ  /\  d  < 
a ) )
1512, 14sylnibr 296 . . . . . 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 2535 . . . . . 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 642 . . . . 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 2529 . . . 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 2669 . . 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 14 . 2  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  ->  { c  e.  QQ  |  c  <  a }  =/=  { c  e.  QQ  | 
c  <  b }
)
21203expa 1151 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 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   class class class wbr 4023   RRcr 8736    < clt 8867   QQcq 10316
This theorem is referenced by:  rpnnen3  27125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317
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