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Theorem kardex 7560
Description: The collection of all sets equinumerous to a set  A and having least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.)
Assertion
Ref Expression
kardex  |-  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) ) }  e.  _V
Distinct variable group:    x, y, A

Proof of Theorem kardex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-rab 2553 . . 3  |-  { x  e.  { z  |  z 
~~  A }  |  A. y  e.  { z  |  z  ~~  A }  ( rank `  x
)  C_  ( rank `  y ) }  =  { x  |  (
x  e.  { z  |  z  ~~  A }  /\  A. y  e. 
{ z  |  z 
~~  A }  ( rank `  x )  C_  ( rank `  y )
) }
2 vex 2792 . . . . . 6  |-  x  e. 
_V
3 breq1 4027 . . . . . 6  |-  ( z  =  x  ->  (
z  ~~  A  <->  x  ~~  A ) )
42, 3elab 2915 . . . . 5  |-  ( x  e.  { z  |  z  ~~  A }  <->  x 
~~  A )
5 breq1 4027 . . . . . 6  |-  ( z  =  y  ->  (
z  ~~  A  <->  y  ~~  A ) )
65ralab 2927 . . . . 5  |-  ( A. y  e.  { z  |  z  ~~  A } 
( rank `  x )  C_  ( rank `  y
)  <->  A. y ( y 
~~  A  ->  ( rank `  x )  C_  ( rank `  y )
) )
74, 6anbi12i 678 . . . 4  |-  ( ( x  e.  { z  |  z  ~~  A }  /\  A. y  e. 
{ z  |  z 
~~  A }  ( rank `  x )  C_  ( rank `  y )
)  <->  ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) ) )
87abbii 2396 . . 3  |-  { x  |  ( x  e. 
{ z  |  z 
~~  A }  /\  A. y  e.  { z  |  z  ~~  A }  ( rank `  x
)  C_  ( rank `  y ) ) }  =  { x  |  ( x  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  x )  C_  ( rank `  y )
) ) }
91, 8eqtri 2304 . 2  |-  { x  e.  { z  |  z 
~~  A }  |  A. y  e.  { z  |  z  ~~  A }  ( rank `  x
)  C_  ( rank `  y ) }  =  { x  |  (
x  ~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) }
10 scottex 7551 . 2  |-  { x  e.  { z  |  z 
~~  A }  |  A. y  e.  { z  |  z  ~~  A }  ( rank `  x
)  C_  ( rank `  y ) }  e.  _V
119, 10eqeltrri 2355 1  |-  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) ) }  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    e. wcel 1685   {cab 2270   A.wral 2544   {crab 2548   _Vcvv 2789    C_ wss 3153   class class class wbr 4024   ` cfv 5221    ~~ cen 6856   rankcrnk 7431
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-reg 7302  ax-inf2 7338
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432  df-rank 7433
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