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Theorem bnj1417 29846
Description: Technical lemma for bnj60 29867. 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 197 . . 3  |-  ( ph  ->  R  FrSe  A )
3 bnj1417.4 . . . . . 6  |-  ( th  <->  (
ph  /\  x  e.  A  /\  ch ) )
4 bnj1418 29845 . . . . . . . . . . 11  |-  ( x  e.  pred ( x ,  A ,  R )  ->  x R x )
54adantl 467 . . . . . . . . . 10  |-  ( ( th  /\  x  e. 
pred ( x ,  A ,  R ) )  ->  x R x )
63, 2bnj835 29566 . . . . . . . . . . . 12  |-  ( th 
->  R  FrSe  A )
7 df-bnj15 29494 . . . . . . . . . . . . 13  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
87simplbi 461 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  R  Fr  A )
96, 8syl 17 . . . . . . . . . . 11  |-  ( th 
->  R  Fr  A
)
10 bnj213 29689 . . . . . . . . . . . 12  |-  pred (
x ,  A ,  R )  C_  A
1110sseli 3460 . . . . . . . . . . 11  |-  ( x  e.  pred ( x ,  A ,  R )  ->  x  e.  A
)
12 frirr 4827 . . . . . . . . . . 11  |-  ( ( R  Fr  A  /\  x  e.  A )  ->  -.  x R x )
139, 11, 12syl2an 479 . . . . . . . . . 10  |-  ( ( th  /\  x  e. 
pred ( x ,  A ,  R ) )  ->  -.  x R x )
145, 13pm2.65da 578 . . . . . . . . 9  |-  ( th 
->  -.  x  e.  pred ( x ,  A ,  R ) )
15 nfv 1751 . . . . . . . . . . . . . 14  |-  F/ y
ph
16 nfv 1751 . . . . . . . . . . . . . 14  |-  F/ y  x  e.  A
17 bnj1417.3 . . . . . . . . . . . . . . . 16  |-  ( ch  <->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
1817bnj1095 29589 . . . . . . . . . . . . . . 15  |-  ( ch 
->  A. y ch )
1918nfi 1670 . . . . . . . . . . . . . 14  |-  F/ y ch
2015, 16, 19nf3an 1986 . . . . . . . . . . . . 13  |-  F/ y ( ph  /\  x  e.  A  /\  ch )
213, 20nfxfr 1692 . . . . . . . . . . . 12  |-  F/ y th
226ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  R  FrSe  A )
23 simplr 760 . . . . . . . . . . . . . . . . 17  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  pred ( x ,  A ,  R ) )
2410, 23sseldi 3462 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  A )
25 simpr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  x  e.  trCl ( y ,  A ,  R ) )
26 bnj1125 29797 . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . 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 29799 . . . . . . . . . . . . . . . . . 18  |-  trCl (
y ,  A ,  R )  C_  A
2928, 25sseldi 3462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  x  e.  A )
30 bnj906 29737 . . . . . . . . . . . . . . . . 17  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R ) 
C_  trCl ( x ,  A ,  R ) )
3122, 29, 30syl2anc 665 . . . . . . . . . . . . . . . 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 3465 . . . . . . . . . . . . . . 15  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  trCl ( x ,  A ,  R ) )
3327, 32sseldd 3465 . . . . . . . . . . . . . 14  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y  e.  trCl ( y ,  A ,  R ) )
3417biimpi 197 . . . . . . . . . . . . . . . . . 18  |-  ( ch 
->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
353, 34bnj837 29568 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  A. y  e.  A  ( y R x  ->  [. y  /  x ]. ps ) )
3635ad2antrr 730 . . . . . . . . . . . . . . . 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 29845 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
3837ad2antlr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  y R x )
39 rsp 2791 . . . . . . . . . . . . . . . 16  |-  ( A. y  e.  A  (
y R x  ->  [. y  /  x ]. ps )  ->  (
y  e.  A  -> 
( y R x  ->  [. y  /  x ]. ps ) ) )
4036, 24, 38, 39syl3c 63 . . . . . . . . . . . . . . 15  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  [. y  /  x ]. ps )
41 vex 3084 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
42 bnj1417.2 . . . . . . . . . . . . . . . . 17  |-  ( ps  <->  -.  x  e.  trCl (
x ,  A ,  R ) )
43 eleq1 2494 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
x  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( x ,  A ,  R ) ) )
44 bnj1318 29830 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  y  ->  trCl (
x ,  A ,  R )  =  trCl ( y ,  A ,  R ) )
4544eleq2d 2492 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  y  ->  (
y  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( y ,  A ,  R ) ) )
4643, 45bitrd 256 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  y  ->  (
x  e.  trCl (
x ,  A ,  R )  <->  y  e.  trCl ( y ,  A ,  R ) ) )
4746notbid 295 . . . . . . . . . . . . . . . . 17  |-  ( x  =  y  ->  ( -.  x  e.  trCl ( x ,  A ,  R )  <->  -.  y  e.  trCl ( y ,  A ,  R ) ) )
4842, 47syl5bb 260 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  ( ps 
<->  -.  y  e.  trCl ( y ,  A ,  R ) ) )
4941, 48sbcie 3334 . . . . . . . . . . . . . . 15  |-  ( [. y  /  x ]. ps  <->  -.  y  e.  trCl (
y ,  A ,  R ) )
5040, 49sylib 199 . . . . . . . . . . . . . 14  |-  ( ( ( th  /\  y  e.  pred ( x ,  A ,  R ) )  /\  x  e. 
trCl ( y ,  A ,  R ) )  ->  -.  y  e.  trCl ( y ,  A ,  R ) )
5133, 50pm2.65da 578 . . . . . . . . . . . . 13  |-  ( ( th  /\  y  e. 
pred ( x ,  A ,  R ) )  ->  -.  x  e.  trCl ( y ,  A ,  R ) )
5251ex 435 . . . . . . . . . . . 12  |-  ( th 
->  ( y  e.  pred ( x ,  A ,  R )  ->  -.  x  e.  trCl ( y ,  A ,  R
) ) )
5321, 52ralrimi 2825 . . . . . . . . . . 11  |-  ( th 
->  A. y  e.  pred  ( x ,  A ,  R )  -.  x  e.  trCl ( y ,  A ,  R ) )
54 ralnex 2871 . . . . . . . . . . 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 199 . . . . . . . . . 10  |-  ( th 
->  -.  E. y  e. 
pred  ( x ,  A ,  R ) x  e.  trCl (
y ,  A ,  R ) )
56 eliun 4301 . . . . . . . . . 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 306 . . . . . . . . 9  |-  ( th 
->  -.  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) )
58 ioran 492 . . . . . . . . 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 668 . . . . . . . 8  |-  ( th 
->  -.  ( x  e. 
pred ( x ,  A ,  R )  \/  x  e.  U_ y  e.  pred  ( x ,  A ,  R
)  trCl ( y ,  A ,  R ) ) )
603simp2bi 1021 . . . . . . . . . . 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 29842 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  trCl ( x ,  A ,  R )  =  B )
636, 60, 62syl2anc 665 . . . . . . . . . 10  |-  ( th 
->  trCl ( x ,  A ,  R )  =  B )
6463eleq2d 2492 . . . . . . . . 9  |-  ( th 
->  ( x  e.  trCl ( x ,  A ,  R )  <->  x  e.  B ) )
6561bnj1138 29596 . . . . . . . . 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 264 . . . . . . . 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 302 . . . . . . 7  |-  ( th 
->  -.  x  e.  trCl ( x ,  A ,  R ) )
6867, 42sylibr 215 . . . . . 6  |-  ( th 
->  ps )
693, 68sylbir 216 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  ch )  ->  ps )
70693exp 1204 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( ch  ->  ps ) ) )
7170ralrimiv 2837 . . 3  |-  ( ph  ->  A. x  e.  A  ( ch  ->  ps )
)
7217bnj1204 29817 . . 3  |-  ( ( R  FrSe  A  /\  A. x  e.  A  ( ch  ->  ps )
)  ->  A. x  e.  A  ps )
732, 71, 72syl2anc 665 . 2  |-  ( ph  ->  A. x  e.  A  ps )
7442ralbii 2856 . 2  |-  ( A. x  e.  A  ps  <->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R
) )
7573, 74sylib 199 1  |-  ( ph  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   [.wsbc 3299    u. cun 3434    C_ wss 3436   U_ciun 4296   class class class wbr 4420    Fr wfr 4806    predc-bnj14 29489    Se w-bnj13 29491    FrSe w-bnj15 29493    trClc-bnj18 29495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-reg 8110  ax-inf2 8149
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-om 6704  df-1o 7187  df-bnj17 29488  df-bnj14 29490  df-bnj13 29492  df-bnj15 29494  df-bnj18 29496  df-bnj19 29498
This theorem is referenced by:  bnj1421  29847
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