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Theorem en1 6944
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.
StepHypRef Expression
1 df1o2 6507 . . . . 5  |-  1o  =  { (/) }
21breq2i 4047 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 6887 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 240 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5501 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5495 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5470 . . . . . . 7  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 15 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5488 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 4166 . . . . . . . . . . 11  |-  (/)  e.  _V
1110fsn2 5714 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 450 . . . . . . . . 9  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 15 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 4922 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 5169 . . . . . . 7  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15syl6eq 2344 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2330 . . . . 5  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
18 fvex 5555 . . . . . 6  |-  ( `' f `  (/) )  e. 
_V
19 sneq 3664 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2019eqeq2d 2307 . . . . . 6  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2118, 20spcev 2888 . . . . 5  |-  ( A  =  { ( `' f `  (/) ) }  ->  E. x  A  =  { x } )
225, 17, 213syl 18 . . . 4  |-  ( f : A -1-1-onto-> { (/) }  ->  E. x  A  =  { x } )
2322exlimiv 1624 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
244, 23sylbi 187 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
25 vex 2804 . . . . 5  |-  x  e. 
_V
2625ensn1 6941 . . . 4  |-  { x }  ~~  1o
27 breq1 4042 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
2826, 27mpbiri 224 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
2928exlimiv 1624 . 2  |-  ( E. x  A  =  {
x }  ->  A  ~~  1o )
3024, 29impbii 180 1  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039   `'ccnv 4704   ran crn 4706   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271   1oc1o 6488    ~~ cen 6876
This theorem is referenced by:  en1b  6945  reuen1  6946  en2  7110  card1  7617  pm54.43  7649  ufildom1  17637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-en 6880
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