Theorem en1 6798

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 6429	. . . . 5 |- 1o = { (/) }
2	1	breq2i 4011	. . . 4 |- ( A ~~ 1o <-> A ~~ { (/) } )
3		bren 6746	. . . 4 |- ( A ~~ { (/) } <-> E. f f : A -1-1-onto-> { (/) } )
4	2, 3	bitri 184	. . 3 |- ( A ~~ 1o <-> E. f f : A -1-1-onto-> { (/) } )
5		f1ocnv 5474	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> `' f : { (/) } -1-1-onto-> A )
6		f1ofo 5468	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } -onto-> A )
7		forn 5441	. . . . . . . 8 |- ( `' f : { (/) } -onto-> A -> ran `' f = A )
8	6, 7	syl 14	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = A )
9		f1of 5461	. . . . . . . . . 10 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } --> A )
10		0ex 4130	. . . . . . . . . . . 12 |- (/) e. _V
11	10	fsn2 5690	. . . . . . . . . . 11 |- ( `' f : { (/) } --> A <-> ( ( `' f ` (/) ) e. A /\ `' f = { <. (/) , ( `' f ` (/) ) >. } ) )
12	11	simprbi 275	. . . . . . . . . 10 |- ( `' f : { (/) } --> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
13	9, 12	syl 14	. . . . . . . . 9 |- ( `' f : { (/) } -1-1-onto-> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
14	13	rneqd 4856	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = ran { <. (/) , ( `' f ` (/) ) >. } )
15	10	rnsnop 5109	. . . . . . . 8 |- ran { <. (/) , ( `' f ` (/) ) >. } = { ( `' f ` (/) ) }
16	14, 15	eqtrdi 2226	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = { ( `' f ` (/) ) } )
17	8, 16	eqtr3d 2212	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> A = { ( `' f ` (/) ) } )
18	5, 17	syl 14	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> A = { ( `' f ` (/) ) } )
19		f1ofn 5462	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> `' f Fn { (/) } )
20	10	snid 3623	. . . . . . 7 |- (/) e. { (/) }
21		funfvex 5532	. . . . . . . 8 |- ( ( Fun `' f /\ (/) e. dom `' f ) -> ( `' f ` (/) ) e. _V )
22	21	funfni 5316	. . . . . . 7 |- ( ( `' f Fn { (/) } /\ (/) e. { (/) } ) -> ( `' f ` (/) ) e. _V )
23	19, 20, 22	sylancl 413	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> ( `' f ` (/) ) e. _V )
24		sneq 3603	. . . . . . . 8 |- ( x = ( `' f ` (/) ) -> { x } = { ( `' f ` (/) ) } )
25	24	eqeq2d 2189	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> ( A = { x } <-> A = { ( `' f ` (/) ) } ) )
26	25	spcegv 2825	. . . . . 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 1598	. . 3 |- ( E. f f : A -1-1-onto-> { (/) } -> E. x A = { x } )
30	4, 29	sylbi 121	. 2 |- ( A ~~ 1o -> E. x A = { x } )
31		vex 2740	. . . . 5 |- x e. _V
32	31	ensn1 6795	. . . 4 |- { x } ~~ 1o
33		breq1 4006	. . . 4 |- ( A = { x } -> ( A ~~ 1o <-> { x } ~~ 1o ) )
34	32, 33	mpbiri 168	. . 3 |- ( A = { x } -> A ~~ 1o )
35	34	exlimiv 1598	. 2 |- ( E. x A = { x } -> A ~~ 1o )
36	30, 35	impbii 126	1 |- ( A ~~ 1o <-> E. x A = { x } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   (/)c0 3422   {csn 3592   <.cop 3595   class class class wbr 4003   `'ccnv 4625   ran crn 4627   Fn wfn 5211   -->wf 5212   -onto->wfo 5214   -1-1-onto->wf1o 5215   ` cfv 5216   1oc1o 6409   ~~ cen 6737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1o 6416  df-en 6740

This theorem is referenced by:  en1bg  6799  reuen1  6800  pm54.43  7188