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Theorem bnj1177 29887
Description: Technical lemma for bnj69 29891. 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
bnj1177.2  |-  ( ps  <->  ( X  e.  B  /\  y  e.  B  /\  y R X ) )
bnj1177.3  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
bnj1177.9  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
bnj1177.13  |-  ( (
ph  /\  ps )  ->  B  C_  A )
bnj1177.17  |-  ( (
ph  /\  ps )  ->  X  e.  A )
Assertion
Ref Expression
bnj1177  |-  ( (
ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e. 
_V ) )

Proof of Theorem bnj1177
StepHypRef Expression
1 bnj1177.9 . . 3  |-  ( (
ph  /\  ps )  ->  R  FrSe  A )
2 df-bnj15 29570 . . . 4  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
32simplbi 467 . . 3  |-  ( R 
FrSe  A  ->  R  Fr  A )
41, 3syl 17 . 2  |-  ( (
ph  /\  ps )  ->  R  Fr  A )
5 bnj1177.3 . . . 4  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
6 bnj1147 29875 . . . . 5  |-  trCl ( X ,  A ,  R )  C_  A
7 ssinss1 3651 . . . . 5  |-  (  trCl ( X ,  A ,  R )  C_  A  ->  (  trCl ( X ,  A ,  R )  i^i  B )  C_  A
)
86, 7ax-mp 5 . . . 4  |-  (  trCl ( X ,  A ,  R )  i^i  B
)  C_  A
95, 8eqsstri 3448 . . 3  |-  C  C_  A
109a1i 11 . 2  |-  ( (
ph  /\  ps )  ->  C  C_  A )
11 bnj1177.17 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  X  e.  A )
12 bnj906 29813 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
131, 11, 12syl2anc 673 . . . . . 6  |-  ( (
ph  /\  ps )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
14 ssrin 3648 . . . . . 6  |-  (  pred ( X ,  A ,  R )  C_  trCl ( X ,  A ,  R )  ->  (  pred ( X ,  A ,  R )  i^i  B
)  C_  (  trCl ( X ,  A ,  R )  i^i  B
) )
1513, 14syl 17 . . . . 5  |-  ( (
ph  /\  ps )  ->  (  pred ( X ,  A ,  R )  i^i  B )  C_  (  trCl ( X ,  A ,  R )  i^i  B
) )
16 bnj1177.13 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  B  C_  A )
17 bnj1177.2 . . . . . . . . . 10  |-  ( ps  <->  ( X  e.  B  /\  y  e.  B  /\  y R X ) )
1817simp2bi 1046 . . . . . . . . 9  |-  ( ps 
->  y  e.  B
)
1918adantl 473 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  y  e.  B )
2016, 19sseldd 3419 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y  e.  A )
2117simp3bi 1047 . . . . . . . 8  |-  ( ps 
->  y R X )
2221adantl 473 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y R X )
23 bnj1152 29879 . . . . . . 7  |-  ( y  e.  pred ( X ,  A ,  R )  <->  ( y  e.  A  /\  y R X ) )
2420, 22, 23sylanbrc 677 . . . . . 6  |-  ( (
ph  /\  ps )  ->  y  e.  pred ( X ,  A ,  R ) )
2524, 19elind 3609 . . . . 5  |-  ( (
ph  /\  ps )  ->  y  e.  (  pred ( X ,  A ,  R )  i^i  B
) )
2615, 25sseldd 3419 . . . 4  |-  ( (
ph  /\  ps )  ->  y  e.  (  trCl ( X ,  A ,  R )  i^i  B
) )
27 ne0i 3728 . . . 4  |-  ( y  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  ->  (  trCl ( X ,  A ,  R )  i^i  B
)  =/=  (/) )
2826, 27syl 17 . . 3  |-  ( (
ph  /\  ps )  ->  (  trCl ( X ,  A ,  R )  i^i  B )  =/=  (/) )
295neeq1i 2707 . . 3  |-  ( C  =/=  (/)  <->  (  trCl ( X ,  A ,  R )  i^i  B
)  =/=  (/) )
3028, 29sylibr 217 . 2  |-  ( (
ph  /\  ps )  ->  C  =/=  (/) )
31 bnj893 29811 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
321, 11, 31syl2anc 673 . . 3  |-  ( (
ph  /\  ps )  ->  trCl ( X ,  A ,  R )  e.  _V )
33 inex1g 4539 . . . 4  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  (  trCl ( X ,  A ,  R )  i^i  B )  e.  _V )
345, 33syl5eqel 2553 . . 3  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  C  e.  _V )
3532, 34syl 17 . 2  |-  ( (
ph  /\  ps )  ->  C  e.  _V )
364, 10, 30, 35bnj951 29659 1  |-  ( (
ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e. 
_V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   class class class wbr 4395    Fr wfr 4795    /\ w-bnj17 29563    predc-bnj14 29565    Se w-bnj13 29567    FrSe w-bnj15 29569    trClc-bnj18 29571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-bnj17 29564  df-bnj14 29566  df-bnj13 29568  df-bnj15 29570  df-bnj18 29572
This theorem is referenced by:  bnj1190  29889
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