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Theorem kardex 8355
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 2782 . . 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 3081 . . . . . 6  |-  x  e. 
_V
3 breq1 4420 . . . . . 6  |-  ( z  =  x  ->  (
z  ~~  A  <->  x  ~~  A ) )
42, 3elab 3215 . . . . 5  |-  ( x  e.  { z  |  z  ~~  A }  <->  x 
~~  A )
5 breq1 4420 . . . . . 6  |-  ( z  =  y  ->  (
z  ~~  A  <->  y  ~~  A ) )
65ralab 3229 . . . . 5  |-  ( A. y  e.  { z  |  z  ~~  A } 
( rank `  x )  C_  ( rank `  y
)  <->  A. y ( y 
~~  A  ->  ( rank `  x )  C_  ( rank `  y )
) )
74, 6anbi12i 701 . . . 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 2554 . . 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 2449 . 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 8346 . 2  |-  { x  e.  { z  |  z 
~~  A }  |  A. y  e.  { z  |  z  ~~  A }  ( rank `  x
)  C_  ( rank `  y ) }  e.  _V
119, 10eqeltrri 2505 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 370   A.wal 1435    e. wcel 1867   {cab 2405   A.wral 2773   {crab 2777   _Vcvv 3078    C_ wss 3433   class class class wbr 4417   ` cfv 5592    ~~ cen 7565   rankcrnk 8224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-reg 8098  ax-inf2 8137
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-r1 8225  df-rank 8226
This theorem is referenced by: (None)
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