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Theorem cardcl 6509
Description: The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
cardcl  |-  ( E. y  e.  On  y  ~~  A  ->  ( card `  A )  e.  On )
Distinct variable group:    y, A

Proof of Theorem cardcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-card 6508 . . . 4  |-  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
)
21a1i 9 . . 3  |-  ( E. y  e.  On  y  ~~  A  ->  card  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  x }
) )
3 breq2 3797 . . . . . 6  |-  ( x  =  A  ->  (
y  ~~  x  <->  y  ~~  A ) )
43rabbidv 2594 . . . . 5  |-  ( x  =  A  ->  { y  e.  On  |  y 
~~  x }  =  { y  e.  On  |  y  ~~  A }
)
54inteqd 3649 . . . 4  |-  ( x  =  A  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  A }
)
65adantl 271 . . 3  |-  ( ( E. y  e.  On  y  ~~  A  /\  x  =  A )  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  A }
)
7 encv 6293 . . . . 5  |-  ( y 
~~  A  ->  (
y  e.  _V  /\  A  e.  _V )
)
87simprd 112 . . . 4  |-  ( y 
~~  A  ->  A  e.  _V )
98rexlimivw 2474 . . 3  |-  ( E. y  e.  On  y  ~~  A  ->  A  e. 
_V )
10 intexrabim 3936 . . 3  |-  ( E. y  e.  On  y  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  _V )
112, 6, 9, 10fvmptd 5285 . 2  |-  ( E. y  e.  On  y  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
12 onintrab2im 4270 . 2  |-  ( E. y  e.  On  y  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  On )
1311, 12eqeltrd 2156 1  |-  ( E. y  e.  On  y  ~~  A  ->  ( card `  A )  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   E.wrex 2350   {crab 2353   _Vcvv 2602   |^|cint 3644   class class class wbr 3793    |-> cmpt 3847   Oncon0 4126   ` cfv 4932    ~~ cen 6285   cardccrd 6507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-en 6288  df-card 6508
This theorem is referenced by: (None)
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