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Theorem bnj1190 29817
Description: Technical lemma for bnj69 29819. 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 1024 . . . . . 6  |-  ( ph  ->  B  C_  A )
32adantr 467 . . . . 5  |-  ( (
ph  /\  ps )  ->  B  C_  A )
4 eqid 2451 . . . . . 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 1023 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
76adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
85simp1bi 1023 . . . . . . . . . 10  |-  ( ps 
->  x  e.  B
)
9 ssel2 3427 . . . . . . . . . 10  |-  ( ( B  C_  A  /\  x  e.  B )  ->  x  e.  A )
102, 8, 9syl2an 480 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  x  e.  A )
115, 4, 7, 3, 10bnj1177 29815 . . . . . . . 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 29808 . . . . . . . 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 29814 . . . . . . 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 240 . . . . . . . 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 240 . . . . . . . 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 29813 . . . . . . 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 29812 . . . . . 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 29811 . . . . . 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 29810 . . . . 5  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  A  ->  ( v R u  ->  -.  v  e.  B ) ) ) )
203, 19bnj1171 29809 . . . 4  |-  E. u A. v ( ( ph  /\ 
ps )  ->  (
u  e.  B  /\  ( v  e.  B  ->  -.  v R u ) ) )
2120bnj1186 29816 . . 3  |-  ( (
ph  /\  ps )  ->  E. u  e.  B  A. v  e.  B  -.  v R u )
2221bnj1185 29605 . 2  |-  ( (
ph  /\  ps )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
2322bnj1185 29605 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 188    /\ wa 371    /\ w3a 985    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    i^i cin 3403    C_ wss 3404   (/)c0 3731   class class class wbr 4402    FrSe w-bnj15 29497    trClc-bnj18 29499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-bnj17 29492  df-bnj14 29494  df-bnj13 29496  df-bnj15 29498  df-bnj18 29500  df-bnj19 29502
This theorem is referenced by:  bnj1189  29818
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