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Theorem bnj1175 28545
Description: Technical lemma for bnj69 28551. 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 28241 . . . . 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 28223 . . . . 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 28533 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
z ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
91, 8bnj835 28300 . . . 4  |-  ( ch 
->  trCl ( z ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
)
10 bnj906 28473 . . . . . 6  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
111, 10bnj836 28301 . . . . 5  |-  ( ch 
->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
12 bnj1152 28539 . . . . . . 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 28302 . . . . 5  |-  ( ch 
->  w  e.  pred ( z ,  A ,  R ) )
1511, 14sseldd 3215 . . . 4  |-  ( ch 
->  w  e.  trCl ( z ,  A ,  R ) )
169, 15sseldd 3215 . . 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 1633    e. wcel 1701    i^i cin 3185    C_ wss 3186   class class class wbr 4060    /\ w-bnj17 28222    predc-bnj14 28224    FrSe w-bnj15 28228    trClc-bnj18 28230
This theorem is referenced by:  bnj1190  28549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-reg 7351  ax-inf2 7387
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-1o 6521  df-bnj17 28223  df-bnj14 28225  df-bnj13 28227  df-bnj15 28229  df-bnj18 28231  df-bnj19 28233
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