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Theorem bnj1190 29354
Description: Technical lemma for bnj69 29356. 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
bnj1190.1  |-  ( ph  <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) ) )
bnj1190.2  |-  ( ps  <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )
Assertion
Ref Expression
bnj1190  |-  ( (
ph  /\  ps )  ->  E. w  e.  B  A. z  e.  B  -.  z R w )
Distinct variable groups:    w, B, x, z    y, B, x, z    w, R, x, z    y, R
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)    A( x, y, z, w)

Proof of Theorem bnj1190
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1190.1 . . . . . . 7  |-  ( ph  <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/=  (/) ) )
21simp2bi 971 . . . . . 6  |-  ( ph  ->  B  C_  A )
32adantr 451 . . . . 5  |-  ( (
ph  /\  ps )  ->  B  C_  A )
4 eqid 2296 . . . . . 6  |-  (  trCl ( x ,  A ,  R )  i^i  B
)  =  (  trCl ( x ,  A ,  R )  i^i  B
)
5 bnj1190.2 . . . . . . . . 9  |-  ( ps  <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )
61simp1bi 970 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
76adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
85simp1bi 970 . . . . . . . . . 10  |-  ( ps 
->  x  e.  B
)
9 ssel2 3188 . . . . . . . . . 10  |-  ( ( B  C_  A  /\  x  e.  B )  ->  x  e.  A )
102, 8, 9syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  x  e.  A )
115, 4, 7, 3, 10bnj1177 29352 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  ( R  Fr  A  /\  (  trCl ( x ,  A ,  R
)  i^i  B )  C_  A  /\  (  trCl ( x ,  A ,  R )  i^i  B
)  =/=  (/)  /\  (  trCl ( x ,  A ,  R )  i^i  B
)  e.  _V )
)
12 bnj1154 29345 . . . . . . . 8  |-  ( ( R  Fr  A  /\  (  trCl ( x ,  A ,  R )  i^i  B )  C_  A  /\  (  trCl (
x ,  A ,  R )  i^i  B
)  =/=  (/)  /\  (  trCl ( x ,  A ,  R )  i^i  B
)  e.  _V )  ->  E. u  e.  ( 
trCl ( x ,  A ,  R )  i^i  B ) A. v  e.  (  trCl ( x ,  A ,  R )  i^i  B
)  -.  v R u )
1311, 12bnj1176 29351 . . . . . . 7  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  (  trCl ( x ,  A ,  R )  i^i  B
)  /\  ( (
( R  FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R  FrSe  A  /\  u  e.  A )  /\  v  e.  A
)  ->  ( v R u  ->  -.  v  e.  (  trCl ( x ,  A ,  R
)  i^i  B )
) ) ) )
14 biid 227 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R  FrSe  A  /\  u  e.  A )  /\  ( v  e.  A  /\  v R u ) )  <->  ( ( R 
FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R 
FrSe  A  /\  u  e.  A )  /\  (
v  e.  A  /\  v R u ) ) )
15 biid 227 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R  FrSe  A  /\  u  e.  A )  /\  v  e.  A
)  <->  ( ( R 
FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R 
FrSe  A  /\  u  e.  A )  /\  v  e.  A ) )
164, 14, 15bnj1175 29350 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R  FrSe  A  /\  u  e.  A )  /\  v  e.  A
)  ->  ( v R u  ->  v  e. 
trCl ( x ,  A ,  R ) ) )
174, 13, 16bnj1174 29349 . . . . . 6  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
( ph  /\  ps  /\  u  e.  (  trCl ( x ,  A ,  R )  i^i  B
) )  /\  (
( ( R  FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R  FrSe  A  /\  u  e.  A )  /\  v  e.  A
)  ->  ( v R u  ->  -.  v  e.  B ) ) ) )
184, 15, 7, 10bnj1173 29348 . . . . . 6  |-  ( (
ph  /\  ps  /\  u  e.  (  trCl ( x ,  A ,  R
)  i^i  B )
)  ->  ( (
( R  FrSe  A  /\  x  e.  A  /\  u  e.  trCl ( x ,  A ,  R ) )  /\  ( R  FrSe  A  /\  u  e.  A )  /\  v  e.  A
)  <->  v  e.  A
) )
194, 17, 18bnj1172 29347 . . . . 5  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  A  ->  ( v R u  ->  -.  v  e.  B ) ) ) )
203, 19bnj1171 29346 . . . 4  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  B  ->  -.  v R u ) ) )
2120bnj1186 29353 . . 3  |-  ( (
ph  /\  ps )  ->  E. u  e.  B  A. v  e.  B  -.  v R u )
2221bnj1185 29142 . 2  |-  ( (
ph  /\  ps )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
2322bnj1185 29142 1  |-  ( (
ph  /\  ps )  ->  E. w  e.  B  A. z  e.  B  -.  z R w )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1189  29355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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