Theorem en1 6302

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)

Assertion

Ref	Expression
en1	|- ( A ~~ 1o <-> E. x A = { x } )

Distinct variable group:   x, A

Proof of Theorem en1

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df1o2 6036	. . . . 5 |- 1o = { (/) }
2	1	breq2i 3793	. . . 4 |- ( A ~~ 1o <-> A ~~ { (/) } )
3		bren 6251	. . . 4 |- ( A ~~ { (/) } <-> E. f f : A -1-1-onto-> { (/) } )
4	2, 3	bitri 182	. . 3 |- ( A ~~ 1o <-> E. f f : A -1-1-onto-> { (/) } )
5		f1ocnv 5159	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> `' f : { (/) } -1-1-onto-> A )
6		f1ofo 5153	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } -onto-> A )
7		forn 5129	. . . . . . . 8 |- ( `' f : { (/) } -onto-> A -> ran `' f = A )
8	6, 7	syl 14	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = A )
9		f1of 5146	. . . . . . . . . 10 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } --> A )
10		0ex 3905	. . . . . . . . . . . 12 |- (/) e. _V
11	10	fsn2 5358	. . . . . . . . . . 11 |- ( `' f : { (/) } --> A <-> ( ( `' f ` (/) ) e. A /\ `' f = { <. (/) , ( `' f ` (/) ) >. } ) )
12	11	simprbi 269	. . . . . . . . . 10 |- ( `' f : { (/) } --> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
13	9, 12	syl 14	. . . . . . . . 9 |- ( `' f : { (/) } -1-1-onto-> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
14	13	rneqd 4581	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = ran { <. (/) , ( `' f ` (/) ) >. } )
15	10	rnsnop 4821	. . . . . . . 8 |- ran { <. (/) , ( `' f ` (/) ) >. } = { ( `' f ` (/) ) }
16	14, 15	syl6eq 2129	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = { ( `' f ` (/) ) } )
17	8, 16	eqtr3d 2115	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> A = { ( `' f ` (/) ) } )
18	5, 17	syl 14	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> A = { ( `' f ` (/) ) } )
19		f1ofn 5147	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> `' f Fn { (/) } )
20	10	snid 3425	. . . . . . 7 |- (/) e. { (/) }
21		funfvex 5212	. . . . . . . 8 |- ( ( Fun `' f /\ (/) e. dom `' f ) -> ( `' f ` (/) ) e. _V )
22	21	funfni 5019	. . . . . . 7 |- ( ( `' f Fn { (/) } /\ (/) e. { (/) } ) -> ( `' f ` (/) ) e. _V )
23	19, 20, 22	sylancl 404	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> ( `' f ` (/) ) e. _V )
24		sneq 3409	. . . . . . . 8 |- ( x = ( `' f ` (/) ) -> { x } = { ( `' f ` (/) ) } )
25	24	eqeq2d 2092	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> ( A = { x } <-> A = { ( `' f ` (/) ) } ) )
26	25	spcegv 2686	. . . . . 6 |- ( ( `' f ` (/) ) e. _V -> ( A = { ( `' f ` (/) ) } -> E. x A = { x } ) )
27	23, 26	syl 14	. . . . 5 |- ( `' f : { (/) } -1-1-onto-> A -> ( A = { ( `' f ` (/) ) } -> E. x A = { x } ) )
28	5, 18, 27	sylc 61	. . . 4 |- ( f : A -1-1-onto-> { (/) } -> E. x A = { x } )
29	28	exlimiv 1529	. . 3 |- ( E. f f : A -1-1-onto-> { (/) } -> E. x A = { x } )
30	4, 29	sylbi 119	. 2 |- ( A ~~ 1o -> E. x A = { x } )
31		vex 2604	. . . . 5 |- x e. _V
32	31	ensn1 6299	. . . 4 |- { x } ~~ 1o
33		breq1 3788	. . . 4 |- ( A = { x } -> ( A ~~ 1o <-> { x } ~~ 1o ) )
34	32, 33	mpbiri 166	. . 3 |- ( A = { x } -> A ~~ 1o )
35	34	exlimiv 1529	. 2 |- ( E. x A = { x } -> A ~~ 1o )
36	30, 35	impbii 124	1 |- ( A ~~ 1o <-> E. x A = { x } )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   (/)c0 3251   {csn 3398   <.cop 3401   class class class wbr 3785   `'ccnv 4362   ran crn 4364   Fn wfn 4917   -->wf 4918   -onto->wfo 4920   -1-1-onto->wf1o 4921   ` cfv 4922   1oc1o 6017   ~~ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-1o 6024  df-en 6245

This theorem is referenced by:  en1bg  6303  reuen1  6304  pm54.43  6459