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Theorem bnj1177 29377
Description: Technical lemma for bnj69 29381. 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 29059 . . . 4  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
32simplbi 448 . . 3  |-  ( R 
FrSe  A  ->  R  Fr  A )
41, 3syl 16 . 2  |-  ( (
ph  /\  ps )  ->  R  Fr  A )
5 bnj1177.3 . . . 4  |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )
6 bnj1147 29365 . . . . 5  |-  trCl ( X ,  A ,  R )  C_  A
7 ssinss1 3571 . . . . 5  |-  (  trCl ( X ,  A ,  R )  C_  A  ->  (  trCl ( X ,  A ,  R )  i^i  B )  C_  A
)
86, 7ax-mp 8 . . . 4  |-  (  trCl ( X ,  A ,  R )  i^i  B
)  C_  A
95, 8eqsstri 3380 . . 3  |-  C  C_  A
109a1i 11 . 2  |-  ( (
ph  /\  ps )  ->  C  C_  A )
11 bnj1177.17 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  X  e.  A )
12 bnj906 29303 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
131, 11, 12syl2anc 644 . . . . . 6  |-  ( (
ph  /\  ps )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
14 ssrin 3568 . . . . . 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 16 . . . . 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 974 . . . . . . . . 9  |-  ( ps 
->  y  e.  B
)
1918adantl 454 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  y  e.  B )
2016, 19sseldd 3351 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y  e.  A )
2117simp3bi 975 . . . . . . . 8  |-  ( ps 
->  y R X )
2221adantl 454 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y R X )
23 bnj1152 29369 . . . . . . 7  |-  ( y  e.  pred ( X ,  A ,  R )  <->  ( y  e.  A  /\  y R X ) )
2420, 22, 23sylanbrc 647 . . . . . 6  |-  ( (
ph  /\  ps )  ->  y  e.  pred ( X ,  A ,  R ) )
2524, 19bnj1153 29166 . . . . 5  |-  ( (
ph  /\  ps )  ->  y  e.  (  pred ( X ,  A ,  R )  i^i  B
) )
2615, 25sseldd 3351 . . . 4  |-  ( (
ph  /\  ps )  ->  y  e.  (  trCl ( X ,  A ,  R )  i^i  B
) )
27 ne0i 3636 . . . 4  |-  ( y  e.  (  trCl ( X ,  A ,  R )  i^i  B
)  ->  (  trCl ( X ,  A ,  R )  i^i  B
)  =/=  (/) )
2826, 27syl 16 . . 3  |-  ( (
ph  /\  ps )  ->  (  trCl ( X ,  A ,  R )  i^i  B )  =/=  (/) )
295neeq1i 2613 . . 3  |-  ( C  =/=  (/)  <->  (  trCl ( X ,  A ,  R )  i^i  B
)  =/=  (/) )
3028, 29sylibr 205 . 2  |-  ( (
ph  /\  ps )  ->  C  =/=  (/) )
31 bnj893 29301 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
321, 11, 31syl2anc 644 . . 3  |-  ( (
ph  /\  ps )  ->  trCl ( X ,  A ,  R )  e.  _V )
33 inex1g 4348 . . . 4  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  (  trCl ( X ,  A ,  R )  i^i  B )  e.  _V )
345, 33syl5eqel 2522 . . 3  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  C  e.  _V )
3532, 34syl 16 . 2  |-  ( (
ph  /\  ps )  ->  C  e.  _V )
364, 10, 30, 35bnj951 29148 1  |-  ( (
ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   class class class wbr 4214    Fr wfr 4540    /\ w-bnj17 29052    predc-bnj14 29054    Se w-bnj13 29056    FrSe w-bnj15 29058    trClc-bnj18 29060
This theorem is referenced by:  bnj1190  29379
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 29053  df-bnj14 29055  df-bnj13 29057  df-bnj15 29059  df-bnj18 29061
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