Theorem en1 6765

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 6397	. . . . 5 |- 1o = { (/) }
2	1	breq2i 3990	. . . 4 |- ( A ~~ 1o <-> A ~~ { (/) } )
3		bren 6713	. . . 4 |- ( A ~~ { (/) } <-> E. f f : A -1-1-onto-> { (/) } )
4	2, 3	bitri 183	. . 3 |- ( A ~~ 1o <-> E. f f : A -1-1-onto-> { (/) } )
5		f1ocnv 5445	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> `' f : { (/) } -1-1-onto-> A )
6		f1ofo 5439	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } -onto-> A )
7		forn 5413	. . . . . . . 8 |- ( `' f : { (/) } -onto-> A -> ran `' f = A )
8	6, 7	syl 14	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = A )
9		f1of 5432	. . . . . . . . . 10 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } --> A )
10		0ex 4109	. . . . . . . . . . . 12 |- (/) e. _V
11	10	fsn2 5659	. . . . . . . . . . 11 |- ( `' f : { (/) } --> A <-> ( ( `' f ` (/) ) e. A /\ `' f = { <. (/) , ( `' f ` (/) ) >. } ) )
12	11	simprbi 273	. . . . . . . . . 10 |- ( `' f : { (/) } --> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
13	9, 12	syl 14	. . . . . . . . 9 |- ( `' f : { (/) } -1-1-onto-> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
14	13	rneqd 4833	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = ran { <. (/) , ( `' f ` (/) ) >. } )
15	10	rnsnop 5084	. . . . . . . 8 |- ran { <. (/) , ( `' f ` (/) ) >. } = { ( `' f ` (/) ) }
16	14, 15	eqtrdi 2215	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = { ( `' f ` (/) ) } )
17	8, 16	eqtr3d 2200	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> A = { ( `' f ` (/) ) } )
18	5, 17	syl 14	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> A = { ( `' f ` (/) ) } )
19		f1ofn 5433	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> `' f Fn { (/) } )
20	10	snid 3607	. . . . . . 7 |- (/) e. { (/) }
21		funfvex 5503	. . . . . . . 8 |- ( ( Fun `' f /\ (/) e. dom `' f ) -> ( `' f ` (/) ) e. _V )
22	21	funfni 5288	. . . . . . 7 |- ( ( `' f Fn { (/) } /\ (/) e. { (/) } ) -> ( `' f ` (/) ) e. _V )
23	19, 20, 22	sylancl 410	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> ( `' f ` (/) ) e. _V )
24		sneq 3587	. . . . . . . 8 |- ( x = ( `' f ` (/) ) -> { x } = { ( `' f ` (/) ) } )
25	24	eqeq2d 2177	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> ( A = { x } <-> A = { ( `' f ` (/) ) } ) )
26	25	spcegv 2814	. . . . . 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 62	. . . 4 |- ( f : A -1-1-onto-> { (/) } -> E. x A = { x } )
29	28	exlimiv 1586	. . 3 |- ( E. f f : A -1-1-onto-> { (/) } -> E. x A = { x } )
30	4, 29	sylbi 120	. 2 |- ( A ~~ 1o -> E. x A = { x } )
31		vex 2729	. . . . 5 |- x e. _V
32	31	ensn1 6762	. . . 4 |- { x } ~~ 1o
33		breq1 3985	. . . 4 |- ( A = { x } -> ( A ~~ 1o <-> { x } ~~ 1o ) )
34	32, 33	mpbiri 167	. . 3 |- ( A = { x } -> A ~~ 1o )
35	34	exlimiv 1586	. 2 |- ( E. x A = { x } -> A ~~ 1o )
36	30, 35	impbii 125	1 |- ( A ~~ 1o <-> E. x A = { x } )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   (/)c0 3409   {csn 3576   <.cop 3579   class class class wbr 3982   `'ccnv 4603   ran crn 4605   Fn wfn 5183   -->wf 5184   -onto->wfo 5186   -1-1-onto->wf1o 5187   ` cfv 5188   1oc1o 6377   ~~ cen 6704

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1o 6384  df-en 6707

This theorem is referenced by:  en1bg  6766  reuen1  6767  pm54.43  7146