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Theorem prub 8613
Description: A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
prub  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( -.  C  e.  A  ->  B  <Q  C ) )

Proof of Theorem prub
StepHypRef Expression
1 eleq1 2344 . . . . . . 7  |-  ( B  =  C  ->  ( B  e.  A  <->  C  e.  A ) )
21biimpcd 217 . . . . . 6  |-  ( B  e.  A  ->  ( B  =  C  ->  C  e.  A ) )
32adantl 454 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( B  =  C  ->  C  e.  A
) )
4 prcdnq 8612 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( C  <Q  B  ->  C  e.  A )
)
53, 4jaod 371 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( ( B  =  C  \/  C  <Q  B )  ->  C  e.  A ) )
65con3d 127 . . 3  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( -.  C  e.  A  ->  -.  ( B  =  C  \/  C  <Q  B ) ) )
76adantr 453 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( -.  C  e.  A  ->  -.  ( B  =  C  \/  C  <Q  B ) ) )
8 elprnq 8610 . . 3  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  B  e.  Q. )
9 ltsonq 8588 . . . 4  |-  <Q  Or  Q.
10 sotric 4339 . . . 4  |-  ( ( 
<Q  Or  Q.  /\  ( B  e.  Q.  /\  C  e.  Q. ) )  -> 
( B  <Q  C  <->  -.  ( B  =  C  \/  C  <Q  B ) ) )
119, 10mpan 653 . . 3  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  <Q  C  <->  -.  ( B  =  C  \/  C  <Q  B ) ) )
128, 11sylan 459 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( B  <Q  C  <->  -.  ( B  =  C  \/  C  <Q  B ) ) )
137, 12sylibrd 227 1  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( -.  C  e.  A  ->  B  <Q  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   class class class wbr 4024    Or wor 4312   Q.cnq 8469    <Q cltq 8475   P.cnp 8476
This theorem is referenced by:  genpnnp  8624  psslinpr  8650  ltexprlem6  8660  ltexprlem7  8661  prlem936  8666  reclem4pr  8669
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-mi 8493  df-lti 8494  df-ltpq 8529  df-enq 8530  df-nq 8531  df-ltnq 8537  df-np 8600
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