Theorem cardcl 7490

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.

Step	Hyp	Ref	Expression
1		df-card 7488	. . . 4 |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
2	1	a1i 9	. . 3 |- ( E. y e. On y ~~ A -> card = ( x e. _V |-> |^| { y e. On | y ~~ x } ) )
3		breq2 4118	. . . . . 6 |- ( x = A -> ( y ~~ x <-> y ~~ A ) )
4	3	rabbidv 2804	. . . . 5 |- ( x = A -> { y e. On | y ~~ x } = { y e. On | y ~~ A } )
5	4	inteqd 3959	. . . 4 |- ( x = A -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ A } )
6	5	adantl 277	. . 3 |- ( ( E. y e. On y ~~ A /\ x = A ) -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ A } )
7		encv 6994	. . . . 5 |- ( y ~~ A -> ( y e. _V /\ A e. _V ) )
8	7	simprd 114	. . . 4 |- ( y ~~ A -> A e. _V )
9	8	rexlimivw 2658	. . 3 |- ( E. y e. On y ~~ A -> A e. _V )
10		intexrabim 4270	. . 3 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. _V )
11	2, 6, 9, 10	fvmptd 5763	. 2 |- ( E. y e. On y ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
12		onintrab2im 4645	. 2 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. On )
13	11, 12	eqeltrd 2311	1 |- ( E. y e. On y ~~ A -> ( card ` A ) e. On )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   E.wrex 2523   {crab 2526   _Vcvv 2815   |^|cint 3954   class class class wbr 4114   |-> cmpt 4176   Oncon0 4489   ` cfv 5357   ~~ cen 6986   cardccrd 7486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-en 6989  df-card 7488

This theorem is referenced by:  ficardon  7498