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Theorem bnj1177 29389
Description: Technical lemma for bnj69 29393. 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 29072 . . . 4  |-  ( R 
FrSe  A  <->  ( R  Fr  A  /\  R  Se  A
) )
32simplbi 458 . . 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 29377 . . . . 5  |-  trCl ( X ,  A ,  R )  C_  A
7 ssinss1 3667 . . . . 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 3472 . . 3  |-  C  C_  A
109a1i 11 . 2  |-  ( (
ph  /\  ps )  ->  C  C_  A )
11 bnj1177.17 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  X  e.  A )
12 bnj906 29315 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
131, 11, 12syl2anc 659 . . . . . 6  |-  ( (
ph  /\  ps )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
14 ssrin 3664 . . . . . 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 1013 . . . . . . . . 9  |-  ( ps 
->  y  e.  B
)
1918adantl 464 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  y  e.  B )
2016, 19sseldd 3443 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y  e.  A )
2117simp3bi 1014 . . . . . . . 8  |-  ( ps 
->  y R X )
2221adantl 464 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  y R X )
23 bnj1152 29381 . . . . . . 7  |-  ( y  e.  pred ( X ,  A ,  R )  <->  ( y  e.  A  /\  y R X ) )
2420, 22, 23sylanbrc 662 . . . . . 6  |-  ( (
ph  /\  ps )  ->  y  e.  pred ( X ,  A ,  R ) )
2524, 19elind 3627 . . . . 5  |-  ( (
ph  /\  ps )  ->  y  e.  (  pred ( X ,  A ,  R )  i^i  B
) )
2615, 25sseldd 3443 . . . 4  |-  ( (
ph  /\  ps )  ->  y  e.  (  trCl ( X ,  A ,  R )  i^i  B
) )
27 ne0i 3744 . . . 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 2688 . . 3  |-  ( C  =/=  (/)  <->  (  trCl ( X ,  A ,  R )  i^i  B
)  =/=  (/) )
3028, 29sylibr 212 . 2  |-  ( (
ph  /\  ps )  ->  C  =/=  (/) )
31 bnj893 29313 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
321, 11, 31syl2anc 659 . . 3  |-  ( (
ph  /\  ps )  ->  trCl ( X ,  A ,  R )  e.  _V )
33 inex1g 4537 . . . 4  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  (  trCl ( X ,  A ,  R )  i^i  B )  e.  _V )
345, 33syl5eqel 2494 . . 3  |-  (  trCl ( X ,  A ,  R )  e.  _V  ->  C  e.  _V )
3532, 34syl 17 . 2  |-  ( (
ph  /\  ps )  ->  C  e.  _V )
364, 10, 30, 35bnj951 29161 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3059    i^i cin 3413    C_ wss 3414   (/)c0 3738   class class class wbr 4395    Fr wfr 4779    /\ w-bnj17 29065    predc-bnj14 29067    Se w-bnj13 29069    FrSe w-bnj15 29071    trClc-bnj18 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-reg 8052  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-om 6684  df-1o 7167  df-bnj17 29066  df-bnj14 29068  df-bnj13 29070  df-bnj15 29072  df-bnj18 29074
This theorem is referenced by:  bnj1190  29391
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