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Theorem kardex 8313
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 2823 . . 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 3116 . . . . . 6  |-  x  e. 
_V
3 breq1 4450 . . . . . 6  |-  ( z  =  x  ->  (
z  ~~  A  <->  x  ~~  A ) )
42, 3elab 3250 . . . . 5  |-  ( x  e.  { z  |  z  ~~  A }  <->  x 
~~  A )
5 breq1 4450 . . . . . 6  |-  ( z  =  y  ->  (
z  ~~  A  <->  y  ~~  A ) )
65ralab 3264 . . . . 5  |-  ( A. y  e.  { z  |  z  ~~  A } 
( rank `  x )  C_  ( rank `  y
)  <->  A. y ( y 
~~  A  ->  ( rank `  x )  C_  ( rank `  y )
) )
74, 6anbi12i 697 . . . 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 2601 . . 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 2496 . 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 8304 . 2  |-  { x  e.  { z  |  z 
~~  A }  |  A. y  e.  { z  |  z  ~~  A }  ( rank `  x
)  C_  ( rank `  y ) }  e.  _V
119, 10eqeltrri 2552 1  |-  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   ` cfv 5588    ~~ cen 7514   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-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: (None)
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