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Theorem bnj1417 29387
Description: Technical lemma for bnj60 29408. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1417.1  |-  ( ph  <->  R 
FrSe  A )
bnj1417.2  |-  ( ps  <->  -.  x  e.  trCl (
x ,  A ,  R ) )
bnj1417.3  |-  ( ch  <->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
bnj1417.4  |-  ( th  <->  (
ph  /\  x  e.  A  /\  ch ) )
bnj1417.5  |-  B  =  (  pred ( x ,  A ,  R )  u.  U_ y  e. 
pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R ) )
Assertion
Ref Expression
bnj1417  |-  ( ph  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
Distinct variable groups:    x, A, y    x, R, y    ph, x, y    ps, y
Allowed substitution hints:    ps( x)    ch( x, y)    th( x, y)    B( x, y)

Proof of Theorem bnj1417
StepHypRef Expression
1 bnj1417.1 . . . 4  |-  ( ph  <->  R 
FrSe  A )
21biimpi 186 . . 3  |-  ( ph  ->  R  FrSe  A )
3 bnj1417.4 . . . . . 6  |-  ( th  <->  (
ph  /\  x  e.  A  /\  ch ) )
4 bnj1418 29386 . . . . . . . . . . 11  |-  ( x  e.  pred ( x ,  A ,  R )  ->  x R x )
54adantl 452 . . . . . . . . . 10  |-  ( ( th  /\  x  e. 
pred ( x ,  A ,  R ) )  ->  x R x )
63, 2bnj835 29105 . . . . . . . . . . . 12  |-  ( th 
->  R  FrSe  A )
7 df-bnj15 29034 . . . . . . . . . . . . 13  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
87simplbi 446 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  R  Fr  A )
96, 8syl 15 . . . . . . . . . . 11  |-  ( th 
->  R  Fr  A
)
10 bnj213 29230 . . . . . . . . . . . 12  |-  pred (
x ,  A ,  R )  C_  A
1110sseli 3189 . . . . . . . . . . 11  |-  ( x  e.  pred ( x ,  A ,  R )  ->  x  e.  A
)
12 frirr 4386 . . . . . . . . . . 11  |-  ( ( R  Fr  A  /\  x  e.  A )  ->  -.  x R x )
139, 11, 12syl2an 463 . . . . . . . . . 10  |-  ( ( th  /\  x  e. 
pred ( x ,  A ,  R ) )  ->  -.  x R x )
145, 13pm2.65da 559 . . . . . . . . 9  |-  ( th 
->  -.  x  e.  pred ( x ,  A ,  R ) )
15 nfv 1609 . . . . . . . . . . . . . 14  |-  F/ y
ph
16 nfv 1609 . . . . . . . . . . . . . 14  |-  F/ y  x  e.  A
17 bnj1417.3 . . . . . . . . . . . . . . . 16  |-  ( ch  <->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
1817bnj1095 29129 . . . . . . . . . . . . . . 15  |-  ( ch 
->  A. y ch )
1918nfi 1541 . . . . . . . . . . . . . 14  |-  F/ y ch
2015, 16, 19nf3an 1786 . . . . . . . . . . . . 13  |-  F/ y ( ph  /\  x  e.  A  /\  ch )
213, 20nfxfr 1560 . . . . . . . . . . . 12  |-  F/ y th
226ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  R  FrSe  A )
23 simplr 731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  pred ( x ,  A ,  R ) )
2410, 23sseldi 3191 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  A )
25 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  x  e.  trCl ( y ,  A ,  R ) )
26 bnj1125 29338 . . . . . . . . . . . . . . . 16  |-  ( ( R  FrSe  A  /\  y  e.  A  /\  x  e.  trCl ( y ,  A ,  R
) )  ->  trCl (
x ,  A ,  R )  C_  trCl (
y ,  A ,  R ) )
2722, 24, 25, 26syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  trCl ( x ,  A ,  R
)  C_  trCl ( y ,  A ,  R
) )
28 bnj1147 29340 . . . . . . . . . . . . . . . . . 18  |-  trCl (
y ,  A ,  R )  C_  A
2928, 25sseldi 3191 . . . . . . . . . . . . . . . . 17  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  x  e.  A )
30 bnj906 29278 . . . . . . . . . . . . . . . . 17  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
3122, 29, 30syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  pred ( x ,  A ,  R
)  C_  trCl ( x ,  A ,  R
) )
3231, 23sseldd 3194 . . . . . . . . . . . . . . 15  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  trCl ( x ,  A ,  R ) )
3327, 32sseldd 3194 . . . . . . . . . . . . . 14  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  trCl ( y ,  A ,  R ) )
3417biimpi 186 . . . . . . . . . . . . . . . . . 18  |-  ( ch 
->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
353, 34bnj837 29107 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
3635ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps )
)
37 bnj1418 29386 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
3837ad2antlr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y R x )
39 rsp 2616 . . . . . . . . . . . . . . . 16  |-  ( A. y  e.  A  (
y R x  ->  [. y  /  x ]. ps )  ->  (
y  e.  A  -> 
( y R x  ->  [. y  /  x ]. ps ) ) )
4036, 24, 38, 39syl3c 57 . . . . . . . . . . . . . . 15  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  [. y  /  x ]. ps )
41 vex 2804 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
42 bnj1417.2 . . . . . . . . . . . . . . . . 17  |-  ( ps  <->  -.  x  e.  trCl (
x ,  A ,  R ) )
43 eleq1 2356 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
x  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( x ,  A ,  R ) ) )
44 bnj1318 29371 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
4544eleq2d 2363 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
y  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( y ,  A ,  R ) ) )
4643, 45bitrd 244 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
x  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( y ,  A ,  R ) ) )
4746notbid 285 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  ( -.  x  e.  trCl ( x ,  A ,  R )  <->  -.  y  e.  trCl ( y ,  A ,  R ) ) )
4842, 47syl5bb 248 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( ps 
<->  -.  y  e.  trCl ( y ,  A ,  R ) ) )
4941, 48sbcie 3038 . . . . . . . . . . . . . . 15  |-  ( [. y  /  x ]. ps  <->  -.  y  e.  trCl (
y ,  A ,  R ) )
5040, 49sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  -.  y  e.  trCl ( y ,  A ,  R ) )
5133, 50pm2.65da 559 . . . . . . . . . . . . 13  |-  ( ( th  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  -.  x  e.  trCl ( y ,  A ,  R ) )
5251ex 423 . . . . . . . . . . . 12  |-  ( th 
->  ( y  e.  pred ( x ,  A ,  R )  ->  -.  x  e.  trCl ( y ,  A ,  R
) ) )
5321, 52ralrimi 2637 . . . . . . . . . . 11  |-  ( th 
->  A. y  e.  pred  ( x ,  A ,  R )  -.  x  e.  trCl ( y ,  A ,  R ) )
54 ralnex 2566 . . . . . . . . . . 11  |-  ( A. y  e.  pred  ( x ,  A ,  R
)  -.  x  e. 
trCl ( y ,  A ,  R )  <->  -.  E. y  e.  pred  ( x ,  A ,  R ) x  e. 
trCl ( y ,  A ,  R ) )
5553, 54sylib 188 . . . . . . . . . 10  |-  ( th 
->  -.  E. y  e. 
pred  ( x ,  A ,  R ) x  e.  trCl (
y ,  A ,  R ) )
56 eliun 3925 . . . . . . . . . 10  |-  ( x  e.  U_ y  e. 
pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R )  <->  E. y  e.  pred  ( x ,  A ,  R ) x  e. 
trCl ( y ,  A ,  R ) )
5755, 56sylnibr 296 . . . . . . . . 9  |-  ( th 
->  -.  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) )
58 ioran 476 . . . . . . . . 9  |-  ( -.  ( x  e.  pred ( x ,  A ,  R )  \/  x  e.  U_ y  e.  pred  ( x ,  A ,  R )  trCl (
y ,  A ,  R ) )  <->  ( -.  x  e.  pred ( x ,  A ,  R
)  /\  -.  x  e.  U_ y  e.  pred  ( x ,  A ,  R )  trCl (
y ,  A ,  R ) ) )
5914, 57, 58sylanbrc 645 . . . . . . . 8  |-  ( th 
->  -.  ( x  e. 
pred ( x ,  A ,  R )  \/  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
603simp2bi 971 . . . . . . . . . . 11  |-  ( th 
->  x  e.  A
)
61 bnj1417.5 . . . . . . . . . . . 12  |-  B  =  (  pred ( x ,  A ,  R )  u.  U_ y  e. 
pred  ( x ,  A ,  R ) 
trCl ( y ,  A ,  R ) )
6261bnj1414 29383 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  =  B )
636, 60, 62syl2anc 642 . . . . . . . . . 10  |-  ( th 
->  trCl ( x ,  A ,  R )  =  B )
6463eleq2d 2363 . . . . . . . . 9  |-  ( th 
->  ( x  e.  trCl ( x ,  A ,  R )  <->  x  e.  B ) )
6561bnj1138 29136 . . . . . . . . 9  |-  ( x  e.  B  <->  ( x  e.  pred ( x ,  A ,  R )  \/  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
6664, 65syl6bb 252 . . . . . . . 8  |-  ( th 
->  ( x  e.  trCl ( x ,  A ,  R )  <->  ( x  e.  pred ( x ,  A ,  R )  \/  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) ) )
6759, 66mtbird 292 . . . . . . 7  |-  ( th 
->  -.  x  e.  trCl ( x ,  A ,  R ) )
6867, 42sylibr 203 . . . . . 6  |-  ( th 
->  ps )
693, 68sylbir 204 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  ch )  ->  ps )
70693exp 1150 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( ch  ->  ps ) ) )
7170ralrimiv 2638 . . 3  |-  ( ph  ->  A. x  e.  A  ( ch  ->  ps )
)
7217bnj1204 29358 . . 3  |-  ( ( R  FrSe  A  /\  A. x  e.  A  ( ch  ->  ps )
)  ->  A. x  e.  A  ps )
732, 71, 72syl2anc 642 . 2  |-  ( ph  ->  A. x  e.  A  ps )
7442ralbii 2580 . 2  |-  ( A. x  e.  A  ps  <->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R
) )
7573, 74sylib 188 1  |-  ( ph  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   [.wsbc 3004    u. cun 3163    C_ wss 3165   U_ciun 3921   class class class wbr 4039    Fr wfr 4365    predc-bnj14 29029    Se w-bnj13 29031    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1421  29388
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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