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Theorem bnj1175 33157
Description: Technical lemma for bnj69 33163. 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 32855 . . . . 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 32837 . . . . 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 273 . . . 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 695 . . . 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 252 . . 3  |-  ( ch  <->  ( th  /\  w R z ) )
8 bnj1125 33145 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
z ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
91, 8bnj835 32914 . . . 4  |-  ( ch 
->  trCl ( z ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
)
10 bnj906 33085 . . . . . 6  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
111, 10bnj836 32915 . . . . 5  |-  ( ch 
->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
12 bnj1152 33151 . . . . . . 7  |-  ( w  e.  pred ( z ,  A ,  R )  <-> 
( w  e.  A  /\  w R z ) )
1312biimpri 206 . . . . . 6  |-  ( ( w  e.  A  /\  w R z )  ->  w  e.  pred ( z ,  A ,  R
) )
141, 13bnj837 32916 . . . . 5  |-  ( ch 
->  w  e.  pred ( z ,  A ,  R ) )
1511, 14sseldd 3505 . . . 4  |-  ( ch 
->  w  e.  trCl ( z ,  A ,  R ) )
169, 15sseldd 3505 . . 3  |-  ( ch 
->  w  e.  trCl ( X ,  A ,  R ) )
177, 16sylbir 213 . 2  |-  ( ( th  /\  w R z )  ->  w  e.  trCl ( X ,  A ,  R )
)
1817ex 434 1  |-  ( th 
->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   class class class wbr 4447    /\ w-bnj17 32836    predc-bnj14 32838    FrSe w-bnj15 32842    trClc-bnj18 32844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-bnj17 32837  df-bnj14 32839  df-bnj13 32841  df-bnj15 32843  df-bnj18 32845  df-bnj19 32847
This theorem is referenced by:  bnj1190  33161
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