MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cplem2 Structured version   Unicode version

Theorem cplem2 8309
Description: -Lemma for the Collection Principle cp 8310. (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 2467 . . 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 2467 . . 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 8308 . 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 8304 . . . 4  |-  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  e.  _V
64, 5iunex 6765 . . 3  |-  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }  e.  _V
7 nfiu1 4355 . . . . 5  |-  F/_ x U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w
) }
87nfeq2 2646 . . . 4  |-  F/ x  y  =  U_ x  e.  A  { z  e.  B  |  A. w  e.  B  ( rank `  z )  C_  ( rank `  w ) }
9 ineq2 3694 . . . . . 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 2744 . . . . 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 316 . . . 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 2898 . . 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 3205 . 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 5 1  |-  E. y A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   U_ciun 4325   ` cfv 5588   rankcrnk 8182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-reg 8019  ax-inf2 8059
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-recs 7043  df-rdg 7077  df-r1 8183  df-rank 8184
This theorem is referenced by:  cp  8310
  Copyright terms: Public domain W3C validator