Theorem isnum2 8771

Description: A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
isnum2	|- ( A e. dom card <-> E. x e. On x ~~ A )

Distinct variable group:   x, A

Proof of Theorem isnum2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardf2 8769	. . . 4 |- card : { y | E. x e. On x ~~ y } --> On
2	1	fdmi 6052	. . 3 |- dom card = { y | E. x e. On x ~~ y }
3	2	eleq2i 2693	. 2 |- ( A e. dom card <-> A e. { y | E. x e. On x ~~ y } )
4		relen 7960	. . . . 5 |- Rel ~~
5	4	brrelex2i 5159	. . . 4 |- ( x ~~ A -> A e. _V )
6	5	rexlimivw 3029	. . 3 |- ( E. x e. On x ~~ A -> A e. _V )
7		breq2 4657	. . . 4 |- ( y = A -> ( x ~~ y <-> x ~~ A ) )
8	7	rexbidv 3052	. . 3 |- ( y = A -> ( E. x e. On x ~~ y <-> E. x e. On x ~~ A ) )
9	6, 8	elab3 3358	. 2 |- ( A e. { y | E. x e. On x ~~ y } <-> E. x e. On x ~~ A )
10	3, 9	bitri 264	1 |- ( A e. dom card <-> E. x e. On x ~~ A )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ~~ cen 7952   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-fun 5890  df-fn 5891  df-f 5892  df-en 7956  df-card 8765

This theorem is referenced by:  isnumi  8772  ennum  8773  xpnum  8777  cardval3  8778  dfac10c  8960  isfin7-2  9218  numth2  9293  inawinalem  9511