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Theorem ltexprlem5 9407
Description: Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  (
y  +Q  x )  e.  B ) }
Assertion
Ref Expression
ltexprlem5  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  C  e.  P. )
Distinct variable groups:    x, y, A    x, B, y    x, C
Allowed substitution hint:    C( y)

Proof of Theorem ltexprlem5
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . . 6  |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  (
y  +Q  x )  e.  B ) }
21ltexprlem1 9403 . . . . 5  |-  ( B  e.  P.  ->  ( A  C.  B  ->  C  =/=  (/) ) )
3 0pss 3852 . . . . 5  |-  ( (/)  C.  C  <->  C  =/=  (/) )
42, 3syl6ibr 227 . . . 4  |-  ( B  e.  P.  ->  ( A  C.  B  ->  (/)  C.  C
) )
54imp 427 . . 3  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  (/)  C.  C )
61ltexprlem2 9404 . . . 4  |-  ( B  e.  P.  ->  C  C. 
Q. )
76adantr 463 . . 3  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  C  C.  Q. )
81ltexprlem3 9405 . . . . . 6  |-  ( B  e.  P.  ->  (
x  e.  C  ->  A. z ( z  <Q  x  ->  z  e.  C
) ) )
91ltexprlem4 9406 . . . . . . 7  |-  ( B  e.  P.  ->  (
x  e.  C  ->  E. z ( z  e.  C  /\  x  <Q  z ) ) )
10 df-rex 2810 . . . . . . 7  |-  ( E. z  e.  C  x 
<Q  z  <->  E. z ( z  e.  C  /\  x  <Q  z ) )
119, 10syl6ibr 227 . . . . . 6  |-  ( B  e.  P.  ->  (
x  e.  C  ->  E. z  e.  C  x  <Q  z ) )
128, 11jcad 531 . . . . 5  |-  ( B  e.  P.  ->  (
x  e.  C  -> 
( A. z ( z  <Q  x  ->  z  e.  C )  /\  E. z  e.  C  x 
<Q  z ) ) )
1312ralrimiv 2866 . . . 4  |-  ( B  e.  P.  ->  A. x  e.  C  ( A. z ( z  <Q  x  ->  z  e.  C
)  /\  E. z  e.  C  x  <Q  z ) )
1413adantr 463 . . 3  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  A. x  e.  C  ( A. z ( z 
<Q  x  ->  z  e.  C )  /\  E. z  e.  C  x  <Q  z ) )
155, 7, 14jca31 532 . 2  |-  ( ( B  e.  P.  /\  A  C.  B )  -> 
( ( (/)  C.  C  /\  C  C.  Q. )  /\  A. x  e.  C  ( A. z ( z 
<Q  x  ->  z  e.  C )  /\  E. z  e.  C  x  <Q  z ) ) )
16 elnp 9354 . 2  |-  ( C  e.  P.  <->  ( ( (/)  C.  C  /\  C  C.  Q. )  /\  A. x  e.  C  ( A. z ( z  <Q  x  ->  z  e.  C
)  /\  E. z  e.  C  x  <Q  z ) ) )
1715, 16sylibr 212 1  |-  ( ( B  e.  P.  /\  A  C.  B )  ->  C  e.  P. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439    =/= wne 2649   A.wral 2804   E.wrex 2805    C. wpss 3462   (/)c0 3783   class class class wbr 4439  (class class class)co 6270   Q.cnq 9219    +Q cplq 9222    <Q cltq 9225   P.cnp 9226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-plpq 9275  df-mpq 9276  df-ltpq 9277  df-enq 9278  df-nq 9279  df-erq 9280  df-plq 9281  df-mq 9282  df-1nq 9283  df-ltnq 9285  df-np 9348
This theorem is referenced by:  ltexprlem6  9408  ltexprlem7  9409  ltexpri  9410
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