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Theorem cplem2 7605
Description: -Lemma for the Collection Principle cp 7606. (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 2316 . . 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 2316 . . 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 7604 . 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 7600 . . . 4  |-  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  e.  _V
64, 5iunex 5812 . . 3  |-  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  e.  _V
7 nfiu1 3970 . . . . 5  |-  F/_ x U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) }
87nfeq2 2463 . . . 4  |-  F/ x  y  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }
9 ineq2 3398 . . . . . 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 2492 . . . . 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 2595 . . 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 2909 . 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 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   {crab 2581   _Vcvv 2822    i^i cin 3185    C_ wss 3186   (/)c0 3489   U_ciun 3942   ` cfv 5292   rankcrnk 7480
This theorem is referenced by:  cp  7606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-reg 7351  ax-inf2 7387
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-recs 6430  df-rdg 6465  df-r1 7481  df-rank 7482
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