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Theorem karden 7560
Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 8168). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 7559 justify the definition of kard. The restriction to least rank prevents the proper class that would result from  { x  |  x  ~~  A }. (Contributed by NM, 18-Dec-2003.)
Hypotheses
Ref Expression
karden.1  |-  A  e. 
_V
karden.2  |-  B  e. 
_V
karden.3  |-  C  =  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) }
karden.4  |-  D  =  { x  |  ( x  ~~  B  /\  A. y ( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) }
Assertion
Ref Expression
karden  |-  ( C  =  D  <->  A  ~~  B )
Distinct variable groups:    x, y, A    x, B, y
Dummy variables  z  w are mutually distinct and distinct from all other variables.
Allowed substitution hints:    C( x, y)    D( x, y)

Proof of Theorem karden
StepHypRef Expression
1 karden.1 . . . . . . . 8  |-  A  e. 
_V
21enref 6889 . . . . . . 7  |-  A  ~~  A
3 breq1 4027 . . . . . . . 8  |-  ( w  =  A  ->  (
w  ~~  A  <->  A  ~~  A ) )
41, 3spcev 2876 . . . . . . 7  |-  ( A 
~~  A  ->  E. w  w  ~~  A )
52, 4ax-mp 10 . . . . . 6  |-  E. w  w  ~~  A
6 abn0 3474 . . . . . 6  |-  ( { w  |  w  ~~  A }  =/=  (/)  <->  E. w  w  ~~  A )
75, 6mpbir 202 . . . . 5  |-  { w  |  w  ~~  A }  =/=  (/)
8 scott0 7551 . . . . . 6  |-  ( { w  |  w  ~~  A }  =  (/)  <->  { z  e.  { w  |  w 
~~  A }  |  A. y  e.  { w  |  w  ~~  A } 
( rank `  z )  C_  ( rank `  y
) }  =  (/) )
98necon3bii 2479 . . . . 5  |-  ( { w  |  w  ~~  A }  =/=  (/)  <->  { z  e.  { w  |  w 
~~  A }  |  A. y  e.  { w  |  w  ~~  A } 
( rank `  z )  C_  ( rank `  y
) }  =/=  (/) )
107, 9mpbi 201 . . . 4  |-  { z  e.  { w  |  w  ~~  A }  |  A. y  e.  {
w  |  w  ~~  A }  ( rank `  z )  C_  ( rank `  y ) }  =/=  (/)
11 rabn0 3475 . . . 4  |-  ( { z  e.  { w  |  w  ~~  A }  |  A. y  e.  {
w  |  w  ~~  A }  ( rank `  z )  C_  ( rank `  y ) }  =/=  (/)  <->  E. z  e.  {
w  |  w  ~~  A } A. y  e. 
{ w  |  w 
~~  A }  ( rank `  z )  C_  ( rank `  y )
)
1210, 11mpbi 201 . . 3  |-  E. z  e.  { w  |  w 
~~  A } A. y  e.  { w  |  w  ~~  A } 
( rank `  z )  C_  ( rank `  y
)
13 vex 2792 . . . . . . . 8  |-  z  e. 
_V
14 breq1 4027 . . . . . . . 8  |-  ( w  =  z  ->  (
w  ~~  A  <->  z  ~~  A ) )
1513, 14elab 2915 . . . . . . 7  |-  ( z  e.  { w  |  w  ~~  A }  <->  z 
~~  A )
16 breq1 4027 . . . . . . . 8  |-  ( w  =  y  ->  (
w  ~~  A  <->  y  ~~  A ) )
1716ralab 2927 . . . . . . 7  |-  ( A. y  e.  { w  |  w  ~~  A } 
( rank `  z )  C_  ( rank `  y
)  <->  A. y ( y 
~~  A  ->  ( rank `  z )  C_  ( rank `  y )
) )
1815, 17anbi12i 680 . . . . . 6  |-  ( ( z  e.  { w  |  w  ~~  A }  /\  A. y  e.  {
w  |  w  ~~  A }  ( rank `  z )  C_  ( rank `  y ) )  <-> 
( z  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  z )  C_  ( rank `  y )
) ) )
19 simpl 445 . . . . . . . . 9  |-  ( ( z  ~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  z
)  C_  ( rank `  y ) ) )  ->  z  ~~  A
)
2019a1i 12 . . . . . . . 8  |-  ( C  =  D  ->  (
( z  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  z )  C_  ( rank `  y )
) )  ->  z  ~~  A ) )
21 karden.3 . . . . . . . . . . . 12  |-  C  =  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) }
22 karden.4 . . . . . . . . . . . 12  |-  D  =  { x  |  ( x  ~~  B  /\  A. y ( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) }
2321, 22eqeq12i 2297 . . . . . . . . . . 11  |-  ( C  =  D  <->  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) ) }  =  { x  |  ( x  ~~  B  /\  A. y ( y 
~~  B  ->  ( rank `  x )  C_  ( rank `  y )
) ) } )
24 abbi 2394 . . . . . . . . . . 11  |-  ( A. x ( ( x 
~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )  <-> 
( x  ~~  B  /\  A. y ( y 
~~  B  ->  ( rank `  x )  C_  ( rank `  y )
) ) )  <->  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) ) }  =  { x  |  ( x  ~~  B  /\  A. y ( y 
~~  B  ->  ( rank `  x )  C_  ( rank `  y )
) ) } )
2523, 24bitr4i 245 . . . . . . . . . 10  |-  ( C  =  D  <->  A. x
( ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) )  <->  ( x  ~~  B  /\  A. y
( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) ) )
26 breq1 4027 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  ~~  A  <->  z  ~~  A ) )
27 fveq2 5485 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  ( rank `  x )  =  ( rank `  z
) )
2827sseq1d 3206 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  z
)  C_  ( rank `  y ) ) )
2928imbi2d 309 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) )  <->  ( y  ~~  A  ->  ( rank `  z )  C_  ( rank `  y ) ) ) )
3029albidv 1612 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  ( A. y ( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) )  <->  A. y
( y  ~~  A  ->  ( rank `  z
)  C_  ( rank `  y ) ) ) )
3126, 30anbi12d 693 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( x  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  x )  C_  ( rank `  y )
) )  <->  ( z  ~~  A  /\  A. y
( y  ~~  A  ->  ( rank `  z
)  C_  ( rank `  y ) ) ) ) )
32 breq1 4027 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  ~~  B  <->  z  ~~  B ) )
3328imbi2d 309 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) )  <->  ( y  ~~  B  ->  ( rank `  z )  C_  ( rank `  y ) ) ) )
3433albidv 1612 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  ( A. y ( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) )  <->  A. y
( y  ~~  B  ->  ( rank `  z
)  C_  ( rank `  y ) ) ) )
3532, 34anbi12d 693 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( x  ~~  B  /\  A. y ( y 
~~  B  ->  ( rank `  x )  C_  ( rank `  y )
) )  <->  ( z  ~~  B  /\  A. y
( y  ~~  B  ->  ( rank `  z
)  C_  ( rank `  y ) ) ) ) )
3631, 35bibi12d 314 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) )  <->  ( x  ~~  B  /\  A. y
( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) )  <->  ( ( z 
~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  z
)  C_  ( rank `  y ) ) )  <-> 
( z  ~~  B  /\  A. y ( y 
~~  B  ->  ( rank `  z )  C_  ( rank `  y )
) ) ) ) )
3736spv 1943 . . . . . . . . . 10  |-  ( A. x ( ( x 
~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) ) )  <-> 
( x  ~~  B  /\  A. y ( y 
~~  B  ->  ( rank `  x )  C_  ( rank `  y )
) ) )  -> 
( ( z  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  z )  C_  ( rank `  y
) ) )  <->  ( z  ~~  B  /\  A. y
( y  ~~  B  ->  ( rank `  z
)  C_  ( rank `  y ) ) ) ) )
3825, 37sylbi 189 . . . . . . . . 9  |-  ( C  =  D  ->  (
( z  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  z )  C_  ( rank `  y )
) )  <->  ( z  ~~  B  /\  A. y
( y  ~~  B  ->  ( rank `  z
)  C_  ( rank `  y ) ) ) ) )
39 simpl 445 . . . . . . . . 9  |-  ( ( z  ~~  B  /\  A. y ( y  ~~  B  ->  ( rank `  z
)  C_  ( rank `  y ) ) )  ->  z  ~~  B
)
4038, 39syl6bi 221 . . . . . . . 8  |-  ( C  =  D  ->  (
( z  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  z )  C_  ( rank `  y )
) )  ->  z  ~~  B ) )
4120, 40jcad 521 . . . . . . 7  |-  ( C  =  D  ->  (
( z  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  z )  C_  ( rank `  y )
) )  ->  (
z  ~~  A  /\  z  ~~  B ) ) )
42 ensym 6905 . . . . . . . 8  |-  ( z 
~~  A  ->  A  ~~  z )
43 entr 6908 . . . . . . . 8  |-  ( ( A  ~~  z  /\  z  ~~  B )  ->  A  ~~  B )
4442, 43sylan 459 . . . . . . 7  |-  ( ( z  ~~  A  /\  z  ~~  B )  ->  A  ~~  B )
4541, 44syl6 31 . . . . . 6  |-  ( C  =  D  ->  (
( z  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  z )  C_  ( rank `  y )
) )  ->  A  ~~  B ) )
4618, 45syl5bi 210 . . . . 5  |-  ( C  =  D  ->  (
( z  e.  {
w  |  w  ~~  A }  /\  A. y  e.  { w  |  w 
~~  A }  ( rank `  z )  C_  ( rank `  y )
)  ->  A  ~~  B ) )
4746exp3a 427 . . . 4  |-  ( C  =  D  ->  (
z  e.  { w  |  w  ~~  A }  ->  ( A. y  e. 
{ w  |  w 
~~  A }  ( rank `  z )  C_  ( rank `  y )  ->  A  ~~  B ) ) )
4847rexlimdv 2667 . . 3  |-  ( C  =  D  ->  ( E. z  e.  { w  |  w  ~~  A } A. y  e.  { w  |  w  ~~  A } 
( rank `  z )  C_  ( rank `  y
)  ->  A  ~~  B ) )
4912, 48mpi 18 . 2  |-  ( C  =  D  ->  A  ~~  B )
50 enen2 6997 . . . . 5  |-  ( A 
~~  B  ->  (
x  ~~  A  <->  x  ~~  B ) )
51 enen2 6997 . . . . . . 7  |-  ( A 
~~  B  ->  (
y  ~~  A  <->  y  ~~  B ) )
5251imbi1d 310 . . . . . 6  |-  ( A 
~~  B  ->  (
( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) )  <->  ( y  ~~  B  ->  ( rank `  x )  C_  ( rank `  y ) ) ) )
5352albidv 1612 . . . . 5  |-  ( A 
~~  B  ->  ( A. y ( y  ~~  A  ->  ( rank `  x
)  C_  ( rank `  y ) )  <->  A. y
( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) )
5450, 53anbi12d 693 . . . 4  |-  ( A 
~~  B  ->  (
( x  ~~  A  /\  A. y ( y 
~~  A  ->  ( rank `  x )  C_  ( rank `  y )
) )  <->  ( x  ~~  B  /\  A. y
( y  ~~  B  ->  ( rank `  x
)  C_  ( rank `  y ) ) ) ) )
5554abbidv 2398 . . 3  |-  ( A 
~~  B  ->  { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  -> 
( rank `  x )  C_  ( rank `  y
) ) ) }  =  { x  |  ( x  ~~  B  /\  A. y ( y 
~~  B  ->  ( rank `  x )  C_  ( rank `  y )
) ) } )
5655, 21, 223eqtr4g 2341 . 2  |-  ( A 
~~  B  ->  C  =  D )
5749, 56impbii 182 1  |-  ( C  =  D  <->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   {cab 2270    =/= wne 2447   A.wral 2544   E.wrex 2545   {crab 2548   _Vcvv 2789    C_ wss 3153   (/)c0 3456   class class class wbr 4024   ` cfv 5221    ~~ cen 6855   rankcrnk 7430
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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-iin 3909  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 6383  df-rdg 6418  df-er 6655  df-en 6859  df-r1 7431  df-rank 7432
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