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Theorem bnj1175 29601
Description: Technical lemma for bnj69 29607. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1175.3  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
bnj1175.4  |-  ( ch  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  ( w  e.  A  /\  w R z ) ) )
bnj1175.5  |-  ( th  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) )
Assertion
Ref Expression
bnj1175  |-  ( th 
->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R ) ) )

Proof of Theorem bnj1175
StepHypRef Expression
1 bnj1175.4 . . . . 5  |-  ( ch  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  ( w  e.  A  /\  w R z ) ) )
2 bnj255 29298 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A  /\  w R z )  <-> 
( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  ( w  e.  A  /\  w R z ) ) )
3 df-bnj17 29280 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A  /\  w R z )  <-> 
( ( ( R 
FrSe  A  /\  X  e.  A  /\  z  e. 
trCl ( X ,  A ,  R )
)  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A )  /\  w R z ) )
41, 2, 33bitr2i 276 . . . 4  |-  ( ch  <->  ( ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  /\  w R
z ) )
5 bnj1175.5 . . . . 5  |-  ( th  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) )
65anbi1i 699 . . . 4  |-  ( ( th  /\  w R z )  <->  ( (
( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  /\  w R
z ) )
74, 6bitr4i 255 . . 3  |-  ( ch  <->  ( th  /\  w R z ) )
8 bnj1125 29589 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
z ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
91, 8bnj835 29358 . . . 4  |-  ( ch 
->  trCl ( z ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
)
10 bnj906 29529 . . . . . 6  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
111, 10bnj836 29359 . . . . 5  |-  ( ch 
->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
12 bnj1152 29595 . . . . . . 7  |-  ( w  e.  pred ( z ,  A ,  R )  <-> 
( w  e.  A  /\  w R z ) )
1312biimpri 209 . . . . . 6  |-  ( ( w  e.  A  /\  w R z )  ->  w  e.  pred ( z ,  A ,  R
) )
141, 13bnj837 29360 . . . . 5  |-  ( ch 
->  w  e.  pred ( z ,  A ,  R ) )
1511, 14sseldd 3471 . . . 4  |-  ( ch 
->  w  e.  trCl ( z ,  A ,  R ) )
169, 15sseldd 3471 . . 3  |-  ( ch 
->  w  e.  trCl ( X ,  A ,  R ) )
177, 16sylbir 216 . 2  |-  ( ( th  /\  w R z )  ->  w  e.  trCl ( X ,  A ,  R )
)
1817ex 435 1  |-  ( th 
->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    i^i cin 3441    C_ wss 3442   class class class wbr 4426    /\ w-bnj17 29279    predc-bnj14 29281    FrSe w-bnj15 29285    trClc-bnj18 29287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7190  df-bnj17 29280  df-bnj14 29282  df-bnj13 29284  df-bnj15 29286  df-bnj18 29288  df-bnj19 29290
This theorem is referenced by:  bnj1190  29605
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