Theorem cardcl 7145

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 7144	. . . 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 3991	. . . . . 6 |- ( x = A -> ( y ~~ x <-> y ~~ A ) )
4	3	rabbidv 2719	. . . . 5 |- ( x = A -> { y e. On | y ~~ x } = { y e. On | y ~~ A } )
5	4	inteqd 3834	. . . 4 |- ( x = A -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ A } )
6	5	adantl 275	. . 3 |- ( ( E. y e. On y ~~ A /\ x = A ) -> |^| { y e. On | y ~~ x } = |^| { y e. On | y ~~ A } )
7		encv 6720	. . . . 5 |- ( y ~~ A -> ( y e. _V /\ A e. _V ) )
8	7	simprd 113	. . . 4 |- ( y ~~ A -> A e. _V )
9	8	rexlimivw 2583	. . 3 |- ( E. y e. On y ~~ A -> A e. _V )
10		intexrabim 4137	. . 3 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. _V )
11	2, 6, 9, 10	fvmptd 5575	. 2 |- ( E. y e. On y ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
12		onintrab2im 4500	. 2 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. On )
13	11, 12	eqeltrd 2247	1 |- ( E. y e. On y ~~ A -> ( card ` A ) e. On )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   _Vcvv 2730   |^|cint 3829   class class class wbr 3987   |-> cmpt 4048   Oncon0 4346   ` cfv 5196   ~~ cen 6712   cardccrd 7143

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-en 6715  df-card 7144

This theorem is referenced by: (None)