Theorem en1 6855

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 6484	. . . . 5 |- 1o = { (/) }
2	1	breq2i 4038	. . . 4 |- ( A ~~ 1o <-> A ~~ { (/) } )
3		bren 6803	. . . 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 5514	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> `' f : { (/) } -1-1-onto-> A )
6		f1ofo 5508	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } -onto-> A )
7		forn 5480	. . . . . . . 8 |- ( `' f : { (/) } -onto-> A -> ran `' f = A )
8	6, 7	syl 14	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = A )
9		f1of 5501	. . . . . . . . . 10 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } --> A )
10		0ex 4157	. . . . . . . . . . . 12 |- (/) e. _V
11	10	fsn2 5733	. . . . . . . . . . 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 4892	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = ran { <. (/) , ( `' f ` (/) ) >. } )
15	10	rnsnop 5147	. . . . . . . 8 |- ran { <. (/) , ( `' f ` (/) ) >. } = { ( `' f ` (/) ) }
16	14, 15	eqtrdi 2242	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = { ( `' f ` (/) ) } )
17	8, 16	eqtr3d 2228	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> A = { ( `' f ` (/) ) } )
18	5, 17	syl 14	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> A = { ( `' f ` (/) ) } )
19		f1ofn 5502	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> `' f Fn { (/) } )
20	10	snid 3650	. . . . . . 7 |- (/) e. { (/) }
21		funfvex 5572	. . . . . . . 8 |- ( ( Fun `' f /\ (/) e. dom `' f ) -> ( `' f ` (/) ) e. _V )
22	21	funfni 5355	. . . . . . 7 |- ( ( `' f Fn { (/) } /\ (/) e. { (/) } ) -> ( `' f ` (/) ) e. _V )
23	19, 20, 22	sylancl 413	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> ( `' f ` (/) ) e. _V )
24		sneq 3630	. . . . . . . 8 |- ( x = ( `' f ` (/) ) -> { x } = { ( `' f ` (/) ) } )
25	24	eqeq2d 2205	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> ( A = { x } <-> A = { ( `' f ` (/) ) } ) )
26	25	spcegv 2849	. . . . . 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 1609	. . 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 2763	. . . . 5 |- x e. _V
32	31	ensn1 6852	. . . 4 |- { x } ~~ 1o
33		breq1 4033	. . . 4 |- ( A = { x } -> ( A ~~ 1o <-> { x } ~~ 1o ) )
34	32, 33	mpbiri 168	. . 3 |- ( A = { x } -> A ~~ 1o )
35	34	exlimiv 1609	. 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 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   (/)c0 3447   {csn 3619   <.cop 3622   class class class wbr 4030   `'ccnv 4659   ran crn 4661   Fn wfn 5250   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255   1oc1o 6464   ~~ cen 6794

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6471  df-en 6797

This theorem is referenced by:  en1bg  6856  reuen1  6857  pm54.43  7252