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Theorem bnj1175 29034
Description: Technical lemma for bnj69 29040. 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 28730 . . . . 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 28712 . . . . 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 264 . . . 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 676 . . . 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 243 . . 3  |-  ( ch  <->  ( th  /\  w R z ) )
8 bnj1125 29022 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
z ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
91, 8bnj835 28789 . . . 4  |-  ( ch 
->  trCl ( z ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
)
10 bnj906 28962 . . . . . 6  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
111, 10bnj836 28790 . . . . 5  |-  ( ch 
->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
12 bnj1152 29028 . . . . . . 7  |-  ( w  e.  pred ( z ,  A ,  R )  <-> 
( w  e.  A  /\  w R z ) )
1312biimpri 197 . . . . . 6  |-  ( ( w  e.  A  /\  w R z )  ->  w  e.  pred ( z ,  A ,  R
) )
141, 13bnj837 28791 . . . . 5  |-  ( ch 
->  w  e.  pred ( z ,  A ,  R ) )
1511, 14sseldd 3181 . . . 4  |-  ( ch 
->  w  e.  trCl ( z ,  A ,  R ) )
169, 15sseldd 3181 . . 3  |-  ( ch 
->  w  e.  trCl ( X ,  A ,  R ) )
177, 16sylbir 204 . 2  |-  ( ( th  /\  w R z )  ->  w  e.  trCl ( X ,  A ,  R )
)
1817ex 423 1  |-  ( th 
->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1190  29038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722
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