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Theorem en1 6516
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 6194 . . . . 5  |-  1o  =  { (/) }
21breq2i 3853 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 6464 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 182 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5266 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5260 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5236 . . . . . . . 8  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 14 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5253 . . . . . . . . . 10  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 3966 . . . . . . . . . . . 12  |-  (/)  e.  _V
1110fsn2 5471 . . . . . . . . . . 11  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 269 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 14 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 4664 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 4911 . . . . . . . 8  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15syl6eq 2136 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2122 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
185, 17syl 14 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  A  =  { ( `' f `
 (/) ) } )
19 f1ofn 5254 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  Fn 
{ (/) } )
2010snid 3475 . . . . . . 7  |-  (/)  e.  { (/)
}
21 funfvex 5322 . . . . . . . 8  |-  ( ( Fun  `' f  /\  (/) 
e.  dom  `' f
)  ->  ( `' f `  (/) )  e. 
_V )
2221funfni 5114 . . . . . . 7  |-  ( ( `' f  Fn  { (/) }  /\  (/)  e.  { (/) } )  ->  ( `' f `  (/) )  e. 
_V )
2319, 20, 22sylancl 404 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ( `' f `
 (/) )  e.  _V )
24 sneq 3457 . . . . . . . 8  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2524eqeq2d 2099 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2625spcegv 2707 . . . . . 6  |-  ( ( `' f `  (/) )  e. 
_V  ->  ( A  =  { ( `' f `
 (/) ) }  ->  E. x  A  =  {
x } ) )
2723, 26syl 14 . . . . 5  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ( A  =  { ( `' f `
 (/) ) }  ->  E. x  A  =  {
x } ) )
285, 18, 27sylc 61 . . . 4  |-  ( f : A -1-1-onto-> { (/) }  ->  E. x  A  =  { x } )
2928exlimiv 1534 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
304, 29sylbi 119 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
31 vex 2622 . . . . 5  |-  x  e. 
_V
3231ensn1 6513 . . . 4  |-  { x }  ~~  1o
33 breq1 3848 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
3432, 33mpbiri 166 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
3534exlimiv 1534 . 2  |-  ( E. x  A  =  {
x }  ->  A  ~~  1o )
3630, 35impbii 124 1  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   (/)c0 3286   {csn 3446   <.cop 3449   class class class wbr 3845   `'ccnv 4437   ran crn 4439    Fn wfn 5010   -->wf 5011   -onto->wfo 5013   -1-1-onto->wf1o 5014   ` cfv 5015   1oc1o 6174    ~~ cen 6455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1o 6181  df-en 6458
This theorem is referenced by:  en1bg  6517  reuen1  6518  pm54.43  6818
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