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Theorem bnj1173 29032
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
bnj1173.3  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
bnj1173.5  |-  ( th  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) )
bnj1173.9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
bnj1173.17  |-  ( (
ph  /\  ps )  ->  X  e.  A )
Assertion
Ref Expression
bnj1173  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( th 
<->  w  e.  A ) )

Proof of Theorem bnj1173
StepHypRef Expression
1 bnj1173.5 . . 3  |-  ( th  <->  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) )
2 3simpc 954 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  ->  ( ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) )
3 bnj1173.9 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
433adant3 975 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  R  FrSe  A )
5 bnj1173.17 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  X  e.  A )
653adant3 975 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  X  e.  A )
7 elin 3358 . . . . . . . . 9  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  <->  ( z  e. 
trCl ( X ,  A ,  R )  /\  z  e.  B
) )
87simplbi 446 . . . . . . . 8  |-  ( z  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  ->  z  e.  trCl ( X ,  A ,  R ) )
9 bnj1173.3 . . . . . . . 8  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
108, 9eleq2s 2375 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  trCl ( X ,  A ,  R )
)
11103ad2ant3 978 . . . . . 6  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  trCl ( X ,  A ,  R )
)
12 pm3.21 435 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  /\  ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R )
) ) ) )
134, 6, 11, 12syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  /\  ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R )
) ) ) )
14 bnj170 28723 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  <->  ( ( ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  /\  ( R  FrSe  A  /\  X  e.  A  /\  z  e. 
trCl ( X ,  A ,  R )
) ) )
1513, 14syl6ibr 218 . . . 4  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A )  ->  (
( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
) ) )
162, 15impbid2 195 . . 3  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
( ( R  FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A
)  <->  ( ( R 
FrSe  A  /\  z  e.  A )  /\  w  e.  A ) ) )
171, 16syl5bb 248 . 2  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( th 
<->  ( ( R  FrSe  A  /\  z  e.  A
)  /\  w  e.  A ) ) )
18 bnj1147 29024 . . . . 5  |-  trCl ( X ,  A ,  R )  C_  A
1918, 11bnj1213 28831 . . . 4  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  z  e.  A )
204, 19jca 518 . . 3  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( R  FrSe  A  /\  z  e.  A ) )
2120biantrurd 494 . 2  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  (
w  e.  A  <->  ( ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) ) )
2217, 21bitr4d 247 1  |-  ( (
ph  /\  ps  /\  z  e.  C )  ->  ( th 
<->  w  e.  A ) )
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    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-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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-rab 2552  df-v 2790  df-sbc 2992  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-tr 4114  df-eprel 4305  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-iota 5219  df-fn 5258  df-fv 5263  df-bnj17 28712  df-bnj14 28714  df-bnj18 28720
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