ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  en1 Unicode version

Theorem en1 6759
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 6391 . . . . 5  |-  1o  =  { (/) }
21breq2i 3987 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 6707 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 183 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5442 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5436 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5410 . . . . . . . 8  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 14 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5429 . . . . . . . . . 10  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 4106 . . . . . . . . . . . 12  |-  (/)  e.  _V
1110fsn2 5656 . . . . . . . . . . 11  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 273 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 14 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 4830 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 5081 . . . . . . . 8  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15eqtrdi 2213 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2199 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
185, 17syl 14 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  A  =  { ( `' f `
 (/) ) } )
19 f1ofn 5430 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  Fn 
{ (/) } )
2010snid 3604 . . . . . . 7  |-  (/)  e.  { (/)
}
21 funfvex 5500 . . . . . . . 8  |-  ( ( Fun  `' f  /\  (/) 
e.  dom  `' f
)  ->  ( `' f `  (/) )  e. 
_V )
2221funfni 5285 . . . . . . 7  |-  ( ( `' f  Fn  { (/) }  /\  (/)  e.  { (/) } )  ->  ( `' f `  (/) )  e. 
_V )
2319, 20, 22sylancl 410 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ( `' f `
 (/) )  e.  _V )
24 sneq 3584 . . . . . . . 8  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2524eqeq2d 2176 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2625spcegv 2812 . . . . . 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 1585 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
304, 29sylbi 120 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
31 vex 2727 . . . . 5  |-  x  e. 
_V
3231ensn1 6756 . . . 4  |-  { x }  ~~  1o
33 breq1 3982 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
3432, 33mpbiri 167 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
3534exlimiv 1585 . 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 1342   E.wex 1479    e. wcel 2135   _Vcvv 2724   (/)c0 3407   {csn 3573   <.cop 3576   class class class wbr 3979   `'ccnv 4600   ran crn 4602    Fn wfn 5180   -->wf 5181   -onto->wfo 5183   -1-1-onto->wf1o 5184   ` cfv 5185   1oc1o 6371    ~~ cen 6698
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-nul 4105  ax-pow 4150  ax-pr 4184  ax-un 4408
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-reu 2449  df-v 2726  df-sbc 2950  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-id 4268  df-suc 4346  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-1o 6378  df-en 6701
This theorem is referenced by:  en1bg  6760  reuen1  6761  pm54.43  7140
  Copyright terms: Public domain W3C validator