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Theorem en1 6310
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 6044 . . . . 5  |-  1o  =  { (/) }
21breq2i 3800 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 6259 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 177 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5167 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5161 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5137 . . . . . . . 8  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 14 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5154 . . . . . . . . . 10  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 3912 . . . . . . . . . . . 12  |-  (/)  e.  _V
1110fsn2 5365 . . . . . . . . . . 11  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 264 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 14 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 4591 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 4829 . . . . . . . 8  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15syl6eq 2104 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2090 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
185, 17syl 14 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  A  =  { ( `' f `
 (/) ) } )
19 f1ofn 5155 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  Fn 
{ (/) } )
2010snid 3430 . . . . . . 7  |-  (/)  e.  { (/)
}
21 funfvex 5220 . . . . . . . 8  |-  ( ( Fun  `' f  /\  (/) 
e.  dom  `' f
)  ->  ( `' f `  (/) )  e. 
_V )
2221funfni 5027 . . . . . . 7  |-  ( ( `' f  Fn  { (/) }  /\  (/)  e.  { (/) } )  ->  ( `' f `  (/) )  e. 
_V )
2319, 20, 22sylancl 398 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ( `' f `
 (/) )  e.  _V )
24 sneq 3414 . . . . . . . 8  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2524eqeq2d 2067 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2625spcegv 2658 . . . . . 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 60 . . . 4  |-  ( f : A -1-1-onto-> { (/) }  ->  E. x  A  =  { x } )
2928exlimiv 1505 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
304, 29sylbi 118 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
31 vex 2577 . . . . 5  |-  x  e. 
_V
3231ensn1 6307 . . . 4  |-  { x }  ~~  1o
33 breq1 3795 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
3432, 33mpbiri 161 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
3534exlimiv 1505 . 2  |-  ( E. x  A  =  {
x }  ->  A  ~~  1o )
3630, 35impbii 121 1  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   (/)c0 3252   {csn 3403   <.cop 3406   class class class wbr 3792   `'ccnv 4372   ran crn 4374    Fn wfn 4925   -->wf 4926   -onto->wfo 4928   -1-1-onto->wf1o 4929   ` cfv 4930   1oc1o 6025    ~~ cen 6250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-1o 6032  df-en 6253
This theorem is referenced by:  en1bg  6311  reuen1  6312
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