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Theorem rpnnen3lem 29520
Description: Lemma for rpnnen3 29521. (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 11272 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  ->  E. d  e.  QQ  ( a  < 
d  /\  d  <  b ) )
2 simp2 989 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  e.  QQ )
3 simp3r 1017 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  <  b
)
4 breq1 4395 . . . . . . . 8  |-  ( c  =  d  ->  (
c  <  b  <->  d  <  b ) )
54elrab 3216 . . . . . . 7  |-  ( d  e.  { c  e.  QQ  |  c  < 
b }  <->  ( d  e.  QQ  /\  d  < 
b ) )
62, 3, 5sylanbrc 664 . . . . . 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 1018 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  a  e.  RR )
8 qre 11061 . . . . . . . . . 10  |-  ( d  e.  QQ  ->  d  e.  RR )
983ad2ant2 1010 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  d  e.  RR )
10 simp3l 1016 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  a  <  d
)
117, 9, 10ltnsymd 9626 . . . . . . . 8  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  -.  d  <  a )
1211intnand 907 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR  /\  a  <  b )  /\  d  e.  QQ  /\  (
a  <  d  /\  d  <  b ) )  ->  -.  ( d  e.  QQ  /\  d  < 
a ) )
13 breq1 4395 . . . . . . . 8  |-  ( c  =  d  ->  (
c  <  a  <->  d  <  a ) )
1413elrab 3216 . . . . . . 7  |-  ( d  e.  { c  e.  QQ  |  c  < 
a }  <->  ( d  e.  QQ  /\  d  < 
a ) )
1512, 14sylnibr 305 . . . . . 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 2777 . . . . . 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 661 . . . . 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 2719 . . . 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 2941 . . 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 1188 1  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  a  <  b
)  ->  { c  e.  QQ  |  c  < 
a }  =/=  {
c  e.  QQ  | 
c  <  b }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799   class class class wbr 4392   RRcr 9384    < clt 9521   QQcq 11056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-q 11057
This theorem is referenced by:  rpnnen3  29521
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