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Theorem bnj1175 29375
Description: Technical lemma for bnj69 29381. 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 29071 . . . . 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 29053 . . . . 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 266 . . . 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 678 . . . 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 245 . . 3  |-  ( ch  <->  ( th  /\  w R z ) )
8 bnj1125 29363 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
z ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
91, 8bnj835 29130 . . . 4  |-  ( ch 
->  trCl ( z ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )
)
10 bnj906 29303 . . . . . 6  |-  ( ( R  FrSe  A  /\  z  e.  A )  ->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
111, 10bnj836 29131 . . . . 5  |-  ( ch 
->  pred ( z ,  A ,  R ) 
C_  trCl ( z ,  A ,  R ) )
12 bnj1152 29369 . . . . . . 7  |-  ( w  e.  pred ( z ,  A ,  R )  <-> 
( w  e.  A  /\  w R z ) )
1312biimpri 199 . . . . . 6  |-  ( ( w  e.  A  /\  w R z )  ->  w  e.  pred ( z ,  A ,  R
) )
141, 13bnj837 29132 . . . . 5  |-  ( ch 
->  w  e.  pred ( z ,  A ,  R ) )
1511, 14sseldd 3351 . . . 4  |-  ( ch 
->  w  e.  trCl ( z ,  A ,  R ) )
169, 15sseldd 3351 . . 3  |-  ( ch 
->  w  e.  trCl ( X ,  A ,  R ) )
177, 16sylbir 206 . 2  |-  ( ( th  /\  w R z )  ->  w  e.  trCl ( X ,  A ,  R )
)
1817ex 425 1  |-  ( th 
->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   class class class wbr 4214    /\ w-bnj17 29052    predc-bnj14 29054    FrSe w-bnj15 29058    trClc-bnj18 29060
This theorem is referenced by:  bnj1190  29379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-bnj17 29053  df-bnj14 29055  df-bnj13 29057  df-bnj15 29059  df-bnj18 29061  df-bnj19 29063
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