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Theorem en1 6659
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 6292 . . . . 5  |-  1o  =  { (/) }
21breq2i 3905 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 6607 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 183 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5346 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5340 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5316 . . . . . . . 8  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 14 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5333 . . . . . . . . . 10  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 4023 . . . . . . . . . . . 12  |-  (/)  e.  _V
1110fsn2 5560 . . . . . . . . . . 11  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 271 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 14 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 4736 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 4987 . . . . . . . 8  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15syl6eq 2164 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2150 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
185, 17syl 14 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  A  =  { ( `' f `
 (/) ) } )
19 f1ofn 5334 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  Fn 
{ (/) } )
2010snid 3524 . . . . . . 7  |-  (/)  e.  { (/)
}
21 funfvex 5404 . . . . . . . 8  |-  ( ( Fun  `' f  /\  (/) 
e.  dom  `' f
)  ->  ( `' f `  (/) )  e. 
_V )
2221funfni 5191 . . . . . . 7  |-  ( ( `' f  Fn  { (/) }  /\  (/)  e.  { (/) } )  ->  ( `' f `  (/) )  e. 
_V )
2319, 20, 22sylancl 407 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ( `' f `
 (/) )  e.  _V )
24 sneq 3506 . . . . . . . 8  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2524eqeq2d 2127 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2625spcegv 2746 . . . . . 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 62 . . . 4  |-  ( f : A -1-1-onto-> { (/) }  ->  E. x  A  =  { x } )
2928exlimiv 1560 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
304, 29sylbi 120 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
31 vex 2661 . . . . 5  |-  x  e. 
_V
3231ensn1 6656 . . . 4  |-  { x }  ~~  1o
33 breq1 3900 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
3432, 33mpbiri 167 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
3534exlimiv 1560 . 2  |-  ( E. x  A  =  {
x }  ->  A  ~~  1o )
3630, 35impbii 125 1  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   _Vcvv 2658   (/)c0 3331   {csn 3495   <.cop 3498   class class class wbr 3897   `'ccnv 4506   ran crn 4508    Fn wfn 5086   -->wf 5087   -onto->wfo 5089   -1-1-onto->wf1o 5090   ` cfv 5091   1oc1o 6272    ~~ cen 6598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1o 6279  df-en 6601
This theorem is referenced by:  en1bg  6660  reuen1  6661  pm54.43  7012
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