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Theorem prub 8831
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 2468 . . . . . . 7  |-  ( B  =  C  ->  ( B  e.  A  <->  C  e.  A ) )
21biimpcd 216 . . . . . 6  |-  ( B  e.  A  ->  ( B  =  C  ->  C  e.  A ) )
32adantl 453 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( B  =  C  ->  C  e.  A
) )
4 prcdnq 8830 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( C  <Q  B  ->  C  e.  A )
)
53, 4jaod 370 . . . 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 452 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( -.  C  e.  A  ->  -.  ( B  =  C  \/  C  <Q  B ) ) )
8 elprnq 8828 . . 3  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  B  e.  Q. )
9 ltsonq 8806 . . . 4  |-  <Q  Or  Q.
10 sotric 4493 . . . 4  |-  ( ( 
<Q  Or  Q.  /\  ( B  e.  Q.  /\  C  e.  Q. ) )  -> 
( B  <Q  C  <->  -.  ( B  =  C  \/  C  <Q  B ) ) )
119, 10mpan 652 . . 3  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  <Q  C  <->  -.  ( B  =  C  \/  C  <Q  B ) ) )
128, 11sylan 458 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( B  <Q  C  <->  -.  ( B  =  C  \/  C  <Q  B ) ) )
137, 12sylibrd 226 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 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4176    Or wor 4466   Q.cnq 8687    <Q cltq 8693   P.cnp 8694
This theorem is referenced by:  genpnnp  8842  psslinpr  8868  ltexprlem6  8878  ltexprlem7  8879  prlem936  8884  reclem4pr  8887
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-recs 6596  df-rdg 6631  df-oadd 6691  df-omul 6692  df-er 6868  df-ni 8709  df-mi 8711  df-lti 8712  df-ltpq 8747  df-enq 8748  df-nq 8749  df-ltnq 8755  df-np 8818
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