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Theorem en1 7177
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 6739 . . . . 5  |-  1o  =  { (/) }
21breq2i 4223 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 7120 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 242 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5690 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5684 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5659 . . . . . . 7  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 16 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5677 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 4342 . . . . . . . . . . 11  |-  (/)  e.  _V
1110fsn2 5911 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 452 . . . . . . . . 9  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 16 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 5100 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 5353 . . . . . . 7  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15syl6eq 2486 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2472 . . . . 5  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
18 fvex 5745 . . . . . 6  |-  ( `' f `  (/) )  e. 
_V
19 sneq 3827 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2019eqeq2d 2449 . . . . . 6  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2118, 20spcev 3045 . . . . 5  |-  ( A  =  { ( `' f `  (/) ) }  ->  E. x  A  =  { x } )
225, 17, 213syl 19 . . . 4  |-  ( f : A -1-1-onto-> { (/) }  ->  E. x  A  =  { x } )
2322exlimiv 1645 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
244, 23sylbi 189 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
25 vex 2961 . . . . 5  |-  x  e. 
_V
2625ensn1 7174 . . . 4  |-  { x }  ~~  1o
27 breq1 4218 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
2826, 27mpbiri 226 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
2928exlimiv 1645 . 2  |-  ( E. x  A  =  {
x }  ->  A  ~~  1o )
3024, 29impbii 182 1  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653    e. wcel 1726   (/)c0 3630   {csn 3816   <.cop 3819   class class class wbr 4215   `'ccnv 4880   ran crn 4882   -->wf 5453   -onto->wfo 5455   -1-1-onto->wf1o 5456   ` cfv 5457   1oc1o 6720    ~~ cen 7109
This theorem is referenced by:  en1b  7178  reuen1  7179  en2  7347  card1  7860  pm54.43  7892  hash1snb  11686  ufildom1  17963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1o 6727  df-en 7113
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