Theorem cardcl 7516

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 7514	. . . 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 4129	. . . . . 6 |- ( x = A -> ( y ~~ x <-> y ~~ A ) )
4	3	rabbidv 2810	. . . . 5 |- ( x = A -> { y e. On | y ~~ x } = { y e. On | y ~~ A } )
5	4	inteqd 3970	. . . 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 7018	. . . . 5 |- ( y ~~ A -> ( y e. _V /\ A e. _V ) )
8	7	simprd 114	. . . 4 |- ( y ~~ A -> A e. _V )
9	8	rexlimivw 2664	. . 3 |- ( E. y e. On y ~~ A -> A e. _V )
10		intexrabim 4284	. . 3 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. _V )
11	2, 6, 9, 10	fvmptd 5780	. 2 |- ( E. y e. On y ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
12		onintrab2im 4660	. 2 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. On )
13	11, 12	eqeltrd 2315	1 |- ( E. y e. On y ~~ A -> ( card ` A ) e. On )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   _Vcvv 2821   |^|cint 3965   class class class wbr 4125   |-> cmpt 4187   Oncon0 4503   ` cfv 5372   ~~ cen 7010   cardccrd 7512

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-en 7013  df-card 7514

This theorem is referenced by:  ficardon  7524