Theorem ween 8858

Description: A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013.)

Assertion

Ref	Expression
ween	|- ( A e. dom card <-> E. r r We A )

Distinct variable group:   A, r

Proof of Theorem ween

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac8b 8854	. 2 |- ( A e. dom card -> E. r r We A )
2		weso 5105	. . . . 5 |- ( r We A -> r Or A )
3		vex 3203	. . . . 5 |- r e. _V
4		soex 7109	. . . . 5 |- ( ( r Or A /\ r e. _V ) -> A e. _V )
5	2, 3, 4	sylancl 694	. . . 4 |- ( r We A -> A e. _V )
6	5	exlimiv 1858	. . 3 |- ( E. r r We A -> A e. _V )
7		unipw 4918	. . . . . 6 |- U. ~P A = A
8		weeq2 5103	. . . . . 6 |- ( U. ~P A = A -> ( r We U. ~P A <-> r We A ) )
9	7, 8	ax-mp 5	. . . . 5 |- ( r We U. ~P A <-> r We A )
10	9	exbii 1774	. . . 4 |- ( E. r r We U. ~P A <-> E. r r We A )
11	10	biimpri 218	. . 3 |- ( E. r r We A -> E. r r We U. ~P A )
12		pwexg 4850	. . . . 5 |- ( A e. _V -> ~P A e. _V )
13		dfac8c 8856	. . . . 5 |- ( ~P A e. _V -> ( E. r r We U. ~P A -> E. f A. x e. ~P A ( x =/= (/) -> ( f ` x ) e. x ) ) )
14	12, 13	syl 17	. . . 4 |- ( A e. _V -> ( E. r r We U. ~P A -> E. f A. x e. ~P A ( x =/= (/) -> ( f ` x ) e. x ) ) )
15		dfac8a 8853	. . . 4 |- ( A e. _V -> ( E. f A. x e. ~P A ( x =/= (/) -> ( f ` x ) e. x ) -> A e. dom card ) )
16	14, 15	syld 47	. . 3 |- ( A e. _V -> ( E. r r We U. ~P A -> A e. dom card ) )
17	6, 11, 16	sylc 65	. 2 |- ( E. r r We A -> A e. dom card )
18	1, 17	impbii 199	1 |- ( A e. dom card <-> E. r r We A )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   Or wor 5034   We wwe 5072   dom cdm 5114   ` cfv 5888   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  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-se 5074  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-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-en 7956  df-card 8765

This theorem is referenced by:  ondomen  8860  dfac10  8959  zorn2lem7  9324  fpwwe  9468  canthnumlem  9470  canthp1lem2  9475  pwfseqlem4a  9483  pwfseqlem4  9484  fin2so  33396