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Theorem cplem2 7560
Description: -Lemma for the Collection Principle cp 7561. (Contributed by NM, 17-Oct-2003.)
Hypothesis
Ref Expression
cplem2.1  |-  A  e. 
_V
Assertion
Ref Expression
cplem2  |-  E. y A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) )
Distinct variable groups:    x, y, A    y, B
Allowed substitution hint:    B( x)

Proof of Theorem cplem2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  =  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }
2 eqid 2283 . . 3  |-  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }
31, 2cplem1 7559 . 2  |-  A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) } )  =/=  (/) )
4 cplem2.1 . . . 4  |-  A  e. 
_V
5 scottex 7555 . . . 4  |-  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  e.  _V
64, 5iunex 5770 . . 3  |-  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  e.  _V
7 nfiu1 3933 . . . . 5  |-  F/_ x U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) }
87nfeq2 2430 . . . 4  |-  F/ x  y  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }
9 ineq2 3364 . . . . . 6  |-  ( y  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  ->  ( B  i^i  y )  =  ( B  i^i  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) } ) )
109neeq1d 2459 . . . . 5  |-  ( y  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  ->  ( ( B  i^i  y )  =/=  (/) 
<->  ( B  i^i  U_ x  e.  A  {
z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) } )  =/=  (/) ) )
1110imbi2d 307 . . . 4  |-  ( y  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  ->  ( ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) )  <->  ( B  =/=  (/)  ->  ( B  i^i  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) } )  =/=  (/) ) ) )
128, 11ralbid 2561 . . 3  |-  ( y  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  ->  ( A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) )  <->  A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) } )  =/=  (/) ) ) )
136, 12spcev 2875 . 2  |-  ( A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) } )  =/=  (/) )  ->  E. y A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) ) )
143, 13ax-mp 8 1  |-  E. y A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   U_ciun 3905   ` cfv 5255   rankcrnk 7435
This theorem is referenced by:  cp  7561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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