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Theorem kardex 7566
Description: The collection of all sets equinumerous to a set  A and having the 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 2554 . . 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 2793 . . . . . 6  |-  x  e. 
_V
3 breq1 4028 . . . . . 6  |-  ( z  =  x  ->  (
z  ~~  A  <->  x  ~~  A ) )
42, 3elab 2916 . . . . 5  |-  ( x  e.  { z  |  z  ~~  A }  <->  x 
~~  A )
5 breq1 4028 . . . . . 6  |-  ( z  =  y  ->  (
z  ~~  A  <->  y  ~~  A ) )
65ralab 2928 . . . . 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 2397 . . 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 2305 . 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 7557 . 2  |-  { x  e.  { z  |  z 
~~  A }  |  A. y  e.  { z  |  z  ~~  A }  ( rank `  x
)  C_  ( rank `  y ) }  e.  _V
119, 10eqeltrri 2356 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 1529    e. wcel 1686   {cab 2271   A.wral 2545   {crab 2549   _Vcvv 2790    C_ wss 3154   class class class wbr 4025   ` cfv 5257    ~~ cen 6862   rankcrnk 7437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438  df-rank 7439
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