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Theorem bnj1177 31994
Description: Technical lemma for bnj69 31998. 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 31678 . . . 4  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
32simplbi 460 . . 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 31982 . . . . 5  |-  trCl ( X ,  A ,  R )  C_  A
7 ssinss1 3576 . . . . 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 3384 . . 3  |-  C  C_  A
109a1i 11 . 2  |-  ( (
ph  /\  ps )  ->  C  C_  A )
11 bnj1177.17 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  X  e.  A )
12 bnj906 31920 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
131, 11, 12syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ps )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
14 ssrin 3573 . . . . . 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 1004 . . . . . . . . 9  |-  ( ps 
->  y  e.  B
)
1918adantl 466 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  y  e.  B )
2016, 19sseldd 3355 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y  e.  A )
2117simp3bi 1005 . . . . . . . 8  |-  ( ps 
->  y R X )
2221adantl 466 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y R X )
23 bnj1152 31986 . . . . . . 7  |-  ( y  e.  pred ( X ,  A ,  R )  <->  ( y  e.  A  /\  y R X ) )
2420, 22, 23sylanbrc 664 . . . . . 6  |-  ( (
ph  /\  ps )  ->  y  e.  pred ( X ,  A ,  R ) )
2524, 19elind 3538 . . . . 5  |-  ( (
ph  /\  ps )  ->  y  e.  (  pred ( X ,  A ,  R )  i^i  B
) )
2615, 25sseldd 3355 . . . 4  |-  ( (
ph  /\  ps )  ->  y  e.  (  trCl ( X ,  A ,  R )  i^i  B
) )
27 ne0i 3641 . . . 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 2616 . . 3  |-  ( C  =/=  (/)  <->  (  trCl ( X ,  A ,  R )  i^i  B
)  =/=  (/) )
3028, 29sylibr 212 . 2  |-  ( (
ph  /\  ps )  ->  C  =/=  (/) )
31 bnj893 31918 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
321, 11, 31syl2anc 661 . . 3  |-  ( (
ph  /\  ps )  ->  trCl ( X ,  A ,  R )  e.  _V )
33 inex1g 4433 . . . 4  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  (  trCl ( X ,  A ,  R )  i^i  B )  e.  _V )
345, 33syl5eqel 2525 . . 3  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  C  e.  _V )
3532, 34syl 16 . 2  |-  ( (
ph  /\  ps )  ->  C  e.  _V )
364, 10, 30, 35bnj951 31766 1  |-  ( (
ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e. 
_V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   _Vcvv 2970    i^i cin 3325    C_ wss 3326   (/)c0 3635   class class class wbr 4290    Fr wfr 4674    /\ w-bnj17 31671    predc-bnj14 31673    Se w-bnj13 31675    FrSe w-bnj15 31677    trClc-bnj18 31679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-reg 7805  ax-inf2 7845
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-om 6475  df-1o 6918  df-bnj17 31672  df-bnj14 31674  df-bnj13 31676  df-bnj15 31678  df-bnj18 31680
This theorem is referenced by:  bnj1190  31996
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