Theorem en1 7076

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 6691	. . . . 5 |- 1o = { (/) }
2	1	breq2i 4133	. . . 4 |- ( A ~~ 1o <-> A ~~ { (/) } )
3		bren 7020	. . . 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 5647	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> `' f : { (/) } -1-1-onto-> A )
6		f1ofo 5641	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } -onto-> A )
7		forn 5613	. . . . . . . 8 |- ( `' f : { (/) } -onto-> A -> ran `' f = A )
8	6, 7	syl 14	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = A )
9		f1of 5634	. . . . . . . . . 10 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } --> A )
10		0ex 4255	. . . . . . . . . . . 12 |- (/) e. _V
11	10	fsn2 5873	. . . . . . . . . . 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 5006	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = ran { <. (/) , ( `' f ` (/) ) >. } )
15	10	rnsnop 5263	. . . . . . . 8 |- ran { <. (/) , ( `' f ` (/) ) >. } = { ( `' f ` (/) ) }
16	14, 15	eqtrdi 2287	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = { ( `' f ` (/) ) } )
17	8, 16	eqtr3d 2273	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> A = { ( `' f ` (/) ) } )
18	5, 17	syl 14	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> A = { ( `' f ` (/) ) } )
19		f1ofn 5635	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> `' f Fn { (/) } )
20	10	snid 3736	. . . . . . 7 |- (/) e. { (/) }
21		funfvex 5707	. . . . . . . 8 |- ( ( Fun `' f /\ (/) e. dom `' f ) -> ( `' f ` (/) ) e. _V )
22	21	funfni 5478	. . . . . . 7 |- ( ( `' f Fn { (/) } /\ (/) e. { (/) } ) -> ( `' f ` (/) ) e. _V )
23	19, 20, 22	sylancl 417	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> ( `' f ` (/) ) e. _V )
24		sneq 3716	. . . . . . . 8 |- ( x = ( `' f ` (/) ) -> { x } = { ( `' f ` (/) ) } )
25	24	eqeq2d 2250	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> ( A = { x } <-> A = { ( `' f ` (/) ) } ) )
26	25	spcegv 2913	. . . . . 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 1651	. . 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 2824	. . . . 5 |- x e. _V
32	31	ensn1 7073	. . . 4 |- { x } ~~ 1o
33		breq1 4128	. . . 4 |- ( A = { x } -> ( A ~~ 1o <-> { x } ~~ 1o ) )
34	32, 33	mpbiri 168	. . 3 |- ( A = { x } -> A ~~ 1o )
35	34	exlimiv 1651	. 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {csn 3705   <.cop 3708   class class class wbr 4125   `'ccnv 4768   ran crn 4770   Fn wfn 5367   -->wf 5368   -onto->wfo 5370   -1-1-onto->wf1o 5371   ` cfv 5372   1oc1o 6670   ~~ cen 7010

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-en 7013

This theorem is referenced by:  en1bg  7077  reuen1  7078  eqsndc  7200  pm54.43  7526  upgrex  16258  vtxdumgrfival  16453  1loopgrvd2fi  16460