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Theorem bnj1190 33018
Description: Technical lemma for bnj69 33020. 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 1007 . . . . . 6  |-  ( ph  ->  B  C_  A )
32adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  B  C_  A )
4 eqid 2460 . . . . . 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 1006 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
76adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
85simp1bi 1006 . . . . . . . . . 10  |-  ( ps 
->  x  e.  B
)
9 ssel2 3492 . . . . . . . . . 10  |-  ( ( B  C_  A  /\  x  e.  B )  ->  x  e.  A )
102, 8, 9syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  x  e.  A )
115, 4, 7, 3, 10bnj1177 33016 . . . . . . . 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 33009 . . . . . . . 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 33015 . . . . . . 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 236 . . . . . . . 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 236 . . . . . . . 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 33014 . . . . . . 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 33013 . . . . . 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 33012 . . . . . 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 33011 . . . . 5  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  A  ->  ( v R u  ->  -.  v  e.  B ) ) ) )
203, 19bnj1171 33010 . . . 4  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  B  ->  -.  v R u ) ) )
2120bnj1186 33017 . . 3  |-  ( (
ph  /\  ps )  ->  E. u  e.  B  A. v  e.  B  -.  v R u )
2221bnj1185 32806 . 2  |-  ( (
ph  /\  ps )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
2322bnj1185 32806 1  |-  ( (
ph  /\  ps )  ->  E. w  e.  B  A. z  e.  B  -.  z R w )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808    i^i cin 3468    C_ wss 3469   (/)c0 3778   class class class wbr 4440    FrSe w-bnj15 32699    trClc-bnj18 32701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-bnj17 32694  df-bnj14 32696  df-bnj13 32698  df-bnj15 32700  df-bnj18 32702  df-bnj19 32704
This theorem is referenced by:  bnj1189  33019
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