MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en1 Structured version   Unicode version

Theorem en1 7376
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 6932 . . . . 5  |-  1o  =  { (/) }
21breq2i 4300 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 7319 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 249 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5653 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5648 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5623 . . . . . . 7  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 16 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5641 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 4422 . . . . . . . . . . 11  |-  (/)  e.  _V
1110fsn2 5883 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 464 . . . . . . . . 9  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 16 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 5067 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 5320 . . . . . . 7  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15syl6eq 2491 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2477 . . . . 5  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
18 fvex 5701 . . . . . 6  |-  ( `' f `  (/) )  e. 
_V
19 sneq 3887 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2019eqeq2d 2454 . . . . . 6  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2118, 20spcev 3064 . . . . 5  |-  ( A  =  { ( `' f `  (/) ) }  ->  E. x  A  =  { x } )
225, 17, 213syl 20 . . . 4  |-  ( f : A -1-1-onto-> { (/) }  ->  E. x  A  =  { x } )
2322exlimiv 1688 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
244, 23sylbi 195 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
25 vex 2975 . . . . 5  |-  x  e. 
_V
2625ensn1 7373 . . . 4  |-  { x }  ~~  1o
27 breq1 4295 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
2826, 27mpbiri 233 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
2928exlimiv 1688 . 2  |-  ( E. x  A  =  {
x }  ->  A  ~~  1o )
3024, 29impbii 188 1  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369   E.wex 1586    e. wcel 1756   (/)c0 3637   {csn 3877   <.cop 3883   class class class wbr 4292   `'ccnv 4839   ran crn 4841   -->wf 5414   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418   1oc1o 6913    ~~ cen 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-1o 6920  df-en 7311
This theorem is referenced by:  en1b  7377  reuen1  7378  en2  7548  card1  8138  pm54.43  8170  hash1snb  12171  ufildom1  19499
  Copyright terms: Public domain W3C validator