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Theorem bnj1417 29472
Description: Technical lemma for bnj60 29493. 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 188 . . 3  |-  ( ph  ->  R  FrSe  A )
3 bnj1417.4 . . . . . 6  |-  ( th  <->  (
ph  /\  x  e.  A  /\  ch ) )
4 bnj1418 29471 . . . . . . . . . . 11  |-  ( x  e.  pred ( x ,  A ,  R )  ->  x R x )
54adantl 454 . . . . . . . . . 10  |-  ( ( th  /\  x  e. 
pred ( x ,  A ,  R ) )  ->  x R x )
63, 2bnj835 29190 . . . . . . . . . . . 12  |-  ( th 
->  R  FrSe  A )
7 df-bnj15 29119 . . . . . . . . . . . . 13  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
87simplbi 448 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  R  Fr  A )
96, 8syl 16 . . . . . . . . . . 11  |-  ( th 
->  R  Fr  A
)
10 bnj213 29315 . . . . . . . . . . . 12  |-  pred (
x ,  A ,  R )  C_  A
1110sseli 3346 . . . . . . . . . . 11  |-  ( x  e.  pred ( x ,  A ,  R )  ->  x  e.  A
)
12 frirr 4561 . . . . . . . . . . 11  |-  ( ( R  Fr  A  /\  x  e.  A )  ->  -.  x R x )
139, 11, 12syl2an 465 . . . . . . . . . 10  |-  ( ( th  /\  x  e. 
pred ( x ,  A ,  R ) )  ->  -.  x R x )
145, 13pm2.65da 561 . . . . . . . . 9  |-  ( th 
->  -.  x  e.  pred ( x ,  A ,  R ) )
15 nfv 1630 . . . . . . . . . . . . . 14  |-  F/ y
ph
16 nfv 1630 . . . . . . . . . . . . . 14  |-  F/ y  x  e.  A
17 bnj1417.3 . . . . . . . . . . . . . . . 16  |-  ( ch  <->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
1817bnj1095 29214 . . . . . . . . . . . . . . 15  |-  ( ch 
->  A. y ch )
1918nfi 1561 . . . . . . . . . . . . . 14  |-  F/ y ch
2015, 16, 19nf3an 1850 . . . . . . . . . . . . 13  |-  F/ y ( ph  /\  x  e.  A  /\  ch )
213, 20nfxfr 1580 . . . . . . . . . . . 12  |-  F/ y th
226ad2antrr 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  R  FrSe  A )
23 simplr 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  pred ( x ,  A ,  R ) )
2410, 23sseldi 3348 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  A )
25 simpr 449 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  x  e.  trCl ( y ,  A ,  R ) )
26 bnj1125 29423 . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . 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 29425 . . . . . . . . . . . . . . . . . 18  |-  trCl (
y ,  A ,  R )  C_  A
2928, 25sseldi 3348 . . . . . . . . . . . . . . . . 17  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  x  e.  A )
30 bnj906 29363 . . . . . . . . . . . . . . . . 17  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
3122, 29, 30syl2anc 644 . . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . . 15  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  trCl ( x ,  A ,  R ) )
3327, 32sseldd 3351 . . . . . . . . . . . . . 14  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  trCl ( y ,  A ,  R ) )
3417biimpi 188 . . . . . . . . . . . . . . . . . 18  |-  ( ch 
->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
353, 34bnj837 29192 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
3635ad2antrr 708 . . . . . . . . . . . . . . . 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 29471 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
3837ad2antlr 709 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y R x )
39 rsp 2768 . . . . . . . . . . . . . . . 16  |-  ( A. y  e.  A  (
y R x  ->  [. y  /  x ]. ps )  ->  (
y  e.  A  -> 
( y R x  ->  [. y  /  x ]. ps ) ) )
4036, 24, 38, 39syl3c 60 . . . . . . . . . . . . . . 15  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  [. y  /  x ]. ps )
41 vex 2961 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
42 bnj1417.2 . . . . . . . . . . . . . . . . 17  |-  ( ps  <->  -.  x  e.  trCl (
x ,  A ,  R ) )
43 eleq1 2498 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
x  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( x ,  A ,  R ) ) )
44 bnj1318 29456 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
4544eleq2d 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
y  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( y ,  A ,  R ) ) )
4643, 45bitrd 246 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
x  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( y ,  A ,  R ) ) )
4746notbid 287 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  ( -.  x  e.  trCl ( x ,  A ,  R )  <->  -.  y  e.  trCl ( y ,  A ,  R ) ) )
4842, 47syl5bb 250 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( ps 
<->  -.  y  e.  trCl ( y ,  A ,  R ) ) )
4941, 48sbcie 3197 . . . . . . . . . . . . . . 15  |-  ( [. y  /  x ]. ps  <->  -.  y  e.  trCl (
y ,  A ,  R ) )
5040, 49sylib 190 . . . . . . . . . . . . . 14  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  -.  y  e.  trCl ( y ,  A ,  R ) )
5133, 50pm2.65da 561 . . . . . . . . . . . . 13  |-  ( ( th  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  -.  x  e.  trCl ( y ,  A ,  R ) )
5251ex 425 . . . . . . . . . . . 12  |-  ( th 
->  ( y  e.  pred ( x ,  A ,  R )  ->  -.  x  e.  trCl ( y ,  A ,  R
) ) )
5321, 52ralrimi 2789 . . . . . . . . . . 11  |-  ( th 
->  A. y  e.  pred  ( x ,  A ,  R )  -.  x  e.  trCl ( y ,  A ,  R ) )
54 ralnex 2717 . . . . . . . . . . 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 190 . . . . . . . . . 10  |-  ( th 
->  -.  E. y  e. 
pred  ( x ,  A ,  R ) x  e.  trCl (
y ,  A ,  R ) )
56 eliun 4099 . . . . . . . . . 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 298 . . . . . . . . 9  |-  ( th 
->  -.  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) )
58 ioran 478 . . . . . . . . 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 647 . . . . . . . 8  |-  ( th 
->  -.  ( x  e. 
pred ( x ,  A ,  R )  \/  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
603simp2bi 974 . . . . . . . . . . 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 29468 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  =  B )
636, 60, 62syl2anc 644 . . . . . . . . . 10  |-  ( th 
->  trCl ( x ,  A ,  R )  =  B )
6463eleq2d 2505 . . . . . . . . 9  |-  ( th 
->  ( x  e.  trCl ( x ,  A ,  R )  <->  x  e.  B ) )
6561bnj1138 29221 . . . . . . . . 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 254 . . . . . . . 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 294 . . . . . . 7  |-  ( th 
->  -.  x  e.  trCl ( x ,  A ,  R ) )
6867, 42sylibr 205 . . . . . 6  |-  ( th 
->  ps )
693, 68sylbir 206 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  ch )  ->  ps )
70693exp 1153 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( ch  ->  ps ) ) )
7170ralrimiv 2790 . . 3  |-  ( ph  ->  A. x  e.  A  ( ch  ->  ps )
)
7217bnj1204 29443 . . 3  |-  ( ( R  FrSe  A  /\  A. x  e.  A  ( ch  ->  ps )
)  ->  A. x  e.  A  ps )
732, 71, 72syl2anc 644 . 2  |-  ( ph  ->  A. x  e.  A  ps )
7442ralbii 2731 . 2  |-  ( A. x  e.  A  ps  <->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R
) )
7573, 74sylib 190 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   [.wsbc 3163    u. cun 3320    C_ wss 3322   U_ciun 4095   class class class wbr 4214    Fr wfr 4540    predc-bnj14 29114    Se w-bnj13 29116    FrSe w-bnj15 29118    trClc-bnj18 29120
This theorem is referenced by:  bnj1421  29473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-bnj17 29113  df-bnj14 29115  df-bnj13 29117  df-bnj15 29119  df-bnj18 29121  df-bnj19 29123
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