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Theorem bnj1190 29304
Description: Technical lemma for bnj69 29306. 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 973 . . . . . 6  |-  ( ph  ->  B  C_  A )
32adantr 452 . . . . 5  |-  ( (
ph  /\  ps )  ->  B  C_  A )
4 eqid 2435 . . . . . 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 972 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
76adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
85simp1bi 972 . . . . . . . . . 10  |-  ( ps 
->  x  e.  B
)
9 ssel2 3335 . . . . . . . . . 10  |-  ( ( B  C_  A  /\  x  e.  B )  ->  x  e.  A )
102, 8, 9syl2an 464 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  x  e.  A )
115, 4, 7, 3, 10bnj1177 29302 . . . . . . . 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 29295 . . . . . . . 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 29301 . . . . . . 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 228 . . . . . . . 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 228 . . . . . . . 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 29300 . . . . . . 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 29299 . . . . . 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 29298 . . . . . 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 29297 . . . . 5  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  A  ->  ( v R u  ->  -.  v  e.  B ) ) ) )
203, 19bnj1171 29296 . . . 4  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  B  ->  -.  v R u ) ) )
2120bnj1186 29303 . . 3  |-  ( (
ph  /\  ps )  ->  E. u  e.  B  A. v  e.  B  -.  v R u )
2221bnj1185 29092 . 2  |-  ( (
ph  /\  ps )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
2322bnj1185 29092 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 177    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204    FrSe w-bnj15 28983    trClc-bnj18 28985
This theorem is referenced by:  bnj1189  29305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-bnj17 28978  df-bnj14 28980  df-bnj13 28982  df-bnj15 28984  df-bnj18 28986  df-bnj19 28988
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