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Theorem bnj1190 32332
Description: Technical lemma for bnj69 32334. 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 1004 . . . . . 6  |-  ( ph  ->  B  C_  A )
32adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  B  C_  A )
4 eqid 2454 . . . . . 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 1003 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
76adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
85simp1bi 1003 . . . . . . . . . 10  |-  ( ps 
->  x  e.  B
)
9 ssel2 3460 . . . . . . . . . 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 32330 . . . . . . . 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 32323 . . . . . . . 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 32329 . . . . . . 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 32328 . . . . . . 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 32327 . . . . . 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 32326 . . . . . 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 32325 . . . . 5  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  A  ->  ( v R u  ->  -.  v  e.  B ) ) ) )
203, 19bnj1171 32324 . . . 4  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  B  ->  -.  v R u ) ) )
2120bnj1186 32331 . . 3  |-  ( (
ph  /\  ps )  ->  E. u  e.  B  A. v  e.  B  -.  v R u )
2221bnj1185 32120 . 2  |-  ( (
ph  /\  ps )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
2322bnj1185 32120 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 965    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    i^i cin 3436    C_ wss 3437   (/)c0 3746   class class class wbr 4401    FrSe w-bnj15 32013    trClc-bnj18 32015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-reg 7919  ax-inf2 7959
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-1o 7031  df-bnj17 32008  df-bnj14 32010  df-bnj13 32012  df-bnj15 32014  df-bnj18 32016  df-bnj19 32018
This theorem is referenced by:  bnj1189  32333
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