Theorem en1 8023

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 7572	. . . . 5 |- 1o = { (/) }
2	1	breq2i 4661	. . . 4 |- ( A ~~ 1o <-> A ~~ { (/) } )
3		bren 7964	. . . 4 |- ( A ~~ { (/) } <-> E. f f : A -1-1-onto-> { (/) } )
4	2, 3	bitri 264	. . 3 |- ( A ~~ 1o <-> E. f f : A -1-1-onto-> { (/) } )
5		f1ocnv 6149	. . . . 5 |- ( f : A -1-1-onto-> { (/) } -> `' f : { (/) } -1-1-onto-> A )
6		f1ofo 6144	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } -onto-> A )
7		forn 6118	. . . . . . 7 |- ( `' f : { (/) } -onto-> A -> ran `' f = A )
8	6, 7	syl 17	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = A )
9		f1of 6137	. . . . . . . . 9 |- ( `' f : { (/) } -1-1-onto-> A -> `' f : { (/) } --> A )
10		0ex 4790	. . . . . . . . . . 11 |- (/) e. _V
11	10	fsn2 6403	. . . . . . . . . 10 |- ( `' f : { (/) } --> A <-> ( ( `' f ` (/) ) e. A /\ `' f = { <. (/) , ( `' f ` (/) ) >. } ) )
12	11	simprbi 480	. . . . . . . . 9 |- ( `' f : { (/) } --> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
13	9, 12	syl 17	. . . . . . . 8 |- ( `' f : { (/) } -1-1-onto-> A -> `' f = { <. (/) , ( `' f ` (/) ) >. } )
14	13	rneqd 5353	. . . . . . 7 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = ran { <. (/) , ( `' f ` (/) ) >. } )
15	10	rnsnop 5616	. . . . . . 7 |- ran { <. (/) , ( `' f ` (/) ) >. } = { ( `' f ` (/) ) }
16	14, 15	syl6eq 2672	. . . . . 6 |- ( `' f : { (/) } -1-1-onto-> A -> ran `' f = { ( `' f ` (/) ) } )
17	8, 16	eqtr3d 2658	. . . . 5 |- ( `' f : { (/) } -1-1-onto-> A -> A = { ( `' f ` (/) ) } )
18		fvex 6201	. . . . . 6 |- ( `' f ` (/) ) e. _V
19		sneq 4187	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> { x } = { ( `' f ` (/) ) } )
20	19	eqeq2d 2632	. . . . . 6 |- ( x = ( `' f ` (/) ) -> ( A = { x } <-> A = { ( `' f ` (/) ) } ) )
21	18, 20	spcev 3300	. . . . 5 |- ( A = { ( `' f ` (/) ) } -> E. x A = { x } )
22	5, 17, 21	3syl 18	. . . 4 |- ( f : A -1-1-onto-> { (/) } -> E. x A = { x } )
23	22	exlimiv 1858	. . 3 |- ( E. f f : A -1-1-onto-> { (/) } -> E. x A = { x } )
24	4, 23	sylbi 207	. 2 |- ( A ~~ 1o -> E. x A = { x } )
25		vex 3203	. . . . 5 |- x e. _V
26	25	ensn1 8020	. . . 4 |- { x } ~~ 1o
27		breq1 4656	. . . 4 |- ( A = { x } -> ( A ~~ 1o <-> { x } ~~ 1o ) )
28	26, 27	mpbiri 248	. . 3 |- ( A = { x } -> A ~~ 1o )
29	28	exlimiv 1858	. 2 |- ( E. x A = { x } -> A ~~ 1o )
30	24, 29	impbii 199	1 |- ( A ~~ 1o <-> E. x A = { x } )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   `'ccnv 5113   ran crn 5115   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   1oc1o 7553   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-en 7956

This theorem is referenced by:  en1b  8024  reuen1  8025  en2  8196  card1  8794  pm54.43  8826  hash1snb  13207  ufildom1  21730