Theorem cardcl 7384

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 7382	. . . 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 4092	. . . . . 6 |- ( x = A -> ( y ~~ x <-> y ~~ A ) )
4	3	rabbidv 2791	. . . . 5 |- ( x = A -> { y e. On | y ~~ x } = { y e. On | y ~~ A } )
5	4	inteqd 3933	. . . 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 6914	. . . . 5 |- ( y ~~ A -> ( y e. _V /\ A e. _V ) )
8	7	simprd 114	. . . 4 |- ( y ~~ A -> A e. _V )
9	8	rexlimivw 2646	. . 3 |- ( E. y e. On y ~~ A -> A e. _V )
10		intexrabim 4243	. . 3 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. _V )
11	2, 6, 9, 10	fvmptd 5727	. 2 |- ( E. y e. On y ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
12		onintrab2im 4616	. 2 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. On )
13	11, 12	eqeltrd 2308	1 |- ( E. y e. On y ~~ A -> ( card ` A ) e. On )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   E.wrex 2511   {crab 2514   _Vcvv 2802   |^|cint 3928   class class class wbr 4088   |-> cmpt 4150   Oncon0 4460   ` cfv 5326   ~~ cen 6906   cardccrd 7380

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-en 6909  df-card 7382

This theorem is referenced by:  ficardon  7392