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Theorem cardcl 7042
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 7041 . . . 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 3933 . . . . . 6  |-  ( x  =  A  ->  (
y  ~~  x  <->  y  ~~  A ) )
43rabbidv 2675 . . . . 5  |-  ( x  =  A  ->  { y  e.  On  |  y 
~~  x }  =  { y  e.  On  |  y  ~~  A }
)
54inteqd 3776 . . . 4  |-  ( x  =  A  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  A }
)
65adantl 275 . . 3  |-  ( ( E. y  e.  On  y  ~~  A  /\  x  =  A )  ->  |^| { y  e.  On  |  y 
~~  x }  =  |^| { y  e.  On  |  y  ~~  A }
)
7 encv 6640 . . . . 5  |-  ( y 
~~  A  ->  (
y  e.  _V  /\  A  e.  _V )
)
87simprd 113 . . . 4  |-  ( y 
~~  A  ->  A  e.  _V )
98rexlimivw 2545 . . 3  |-  ( E. y  e.  On  y  ~~  A  ->  A  e. 
_V )
10 intexrabim 4078 . . 3  |-  ( E. y  e.  On  y  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  _V )
112, 6, 9, 10fvmptd 5502 . 2  |-  ( E. y  e.  On  y  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
12 onintrab2im 4434 . 2  |-  ( E. y  e.  On  y  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  On )
1311, 12eqeltrd 2216 1  |-  ( E. y  e.  On  y  ~~  A  ->  ( card `  A )  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   E.wrex 2417   {crab 2420   _Vcvv 2686   |^|cint 3771   class class class wbr 3929    |-> cmpt 3989   Oncon0 4285   ` cfv 5123    ~~ cen 6632   cardccrd 7040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-en 6635  df-card 7041
This theorem is referenced by: (None)
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