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Theorem bnj1177 29808
Description: Technical lemma for bnj69 29812. 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 29491 . . . 4  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
32simplbi 462 . . 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 29796 . . . . 5  |-  trCl ( X ,  A ,  R )  C_  A
7 ssinss1 3659 . . . . 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 3461 . . 3  |-  C  C_  A
109a1i 11 . 2  |-  ( (
ph  /\  ps )  ->  C  C_  A )
11 bnj1177.17 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  X  e.  A )
12 bnj906 29734 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
131, 11, 12syl2anc 666 . . . . . 6  |-  ( (
ph  /\  ps )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
14 ssrin 3656 . . . . . 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 1023 . . . . . . . . 9  |-  ( ps 
->  y  e.  B
)
1918adantl 468 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  y  e.  B )
2016, 19sseldd 3432 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y  e.  A )
2117simp3bi 1024 . . . . . . . 8  |-  ( ps 
->  y R X )
2221adantl 468 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y R X )
23 bnj1152 29800 . . . . . . 7  |-  ( y  e.  pred ( X ,  A ,  R )  <->  ( y  e.  A  /\  y R X ) )
2420, 22, 23sylanbrc 669 . . . . . 6  |-  ( (
ph  /\  ps )  ->  y  e.  pred ( X ,  A ,  R ) )
2524, 19elind 3617 . . . . 5  |-  ( (
ph  /\  ps )  ->  y  e.  (  pred ( X ,  A ,  R )  i^i  B
) )
2615, 25sseldd 3432 . . . 4  |-  ( (
ph  /\  ps )  ->  y  e.  (  trCl ( X ,  A ,  R )  i^i  B
) )
27 ne0i 3736 . . . 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 2687 . . 3  |-  ( C  =/=  (/)  <->  (  trCl ( X ,  A ,  R )  i^i  B
)  =/=  (/) )
3028, 29sylibr 216 . 2  |-  ( (
ph  /\  ps )  ->  C  =/=  (/) )
31 bnj893 29732 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
321, 11, 31syl2anc 666 . . 3  |-  ( (
ph  /\  ps )  ->  trCl ( X ,  A ,  R )  e.  _V )
33 inex1g 4545 . . . 4  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  (  trCl ( X ,  A ,  R )  i^i  B )  e.  _V )
345, 33syl5eqel 2532 . . 3  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  C  e.  _V )
3532, 34syl 17 . 2  |-  ( (
ph  /\  ps )  ->  C  e.  _V )
364, 10, 30, 35bnj951 29580 1  |-  ( (
ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e. 
_V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   _Vcvv 3044    i^i cin 3402    C_ wss 3403   (/)c0 3730   class class class wbr 4401    Fr wfr 4789    /\ w-bnj17 29484    predc-bnj14 29486    Se w-bnj13 29488    FrSe w-bnj15 29490    trClc-bnj18 29492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-reg 8104  ax-inf2 8143
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-1o 7179  df-bnj17 29485  df-bnj14 29487  df-bnj13 29489  df-bnj15 29491  df-bnj18 29493
This theorem is referenced by:  bnj1190  29810
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