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Theorem en1 7165
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 6727 . . . . 5  |-  1o  =  { (/) }
21breq2i 4212 . . . 4  |-  ( A 
~~  1o  <->  A  ~~  { (/) } )
3 bren 7108 . . . 4  |-  ( A 
~~  { (/) }  <->  E. f 
f : A -1-1-onto-> { (/) } )
42, 3bitri 241 . . 3  |-  ( A 
~~  1o  <->  E. f  f : A -1-1-onto-> { (/) } )
5 f1ocnv 5678 . . . . 5  |-  ( f : A -1-1-onto-> { (/) }  ->  `' f : { (/) } -1-1-onto-> A )
6 f1ofo 5672 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } -onto-> A )
7 forn 5647 . . . . . . 7  |-  ( `' f : { (/) }
-onto-> A  ->  ran  `' f  =  A )
86, 7syl 16 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  A )
9 f1of 5665 . . . . . . . . 9  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f : { (/) } --> A )
10 0ex 4331 . . . . . . . . . . 11  |-  (/)  e.  _V
1110fsn2 5899 . . . . . . . . . 10  |-  ( `' f : { (/) } --> A  <->  ( ( `' f `  (/) )  e.  A  /\  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } ) )
1211simprbi 451 . . . . . . . . 9  |-  ( `' f : { (/) } --> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
139, 12syl 16 . . . . . . . 8  |-  ( `' f : { (/) } -1-1-onto-> A  ->  `' f  =  { <. (/) ,  ( `' f `  (/) ) >. } )
1413rneqd 5088 . . . . . . 7  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  ran  { <. (/) ,  ( `' f `  (/) ) >. } )
1510rnsnop 5341 . . . . . . 7  |-  ran  { <.
(/) ,  ( `' f `  (/) ) >. }  =  { ( `' f `  (/) ) }
1614, 15syl6eq 2483 . . . . . 6  |-  ( `' f : { (/) } -1-1-onto-> A  ->  ran  `' f  =  { ( `' f `
 (/) ) } )
178, 16eqtr3d 2469 . . . . 5  |-  ( `' f : { (/) } -1-1-onto-> A  ->  A  =  {
( `' f `  (/) ) } )
18 fvex 5733 . . . . . 6  |-  ( `' f `  (/) )  e. 
_V
19 sneq 3817 . . . . . . 7  |-  ( x  =  ( `' f `
 (/) )  ->  { x }  =  { ( `' f `  (/) ) } )
2019eqeq2d 2446 . . . . . 6  |-  ( x  =  ( `' f `
 (/) )  ->  ( A  =  { x } 
<->  A  =  { ( `' f `  (/) ) } ) )
2118, 20spcev 3035 . . . . 5  |-  ( A  =  { ( `' f `  (/) ) }  ->  E. x  A  =  { x } )
225, 17, 213syl 19 . . . 4  |-  ( f : A -1-1-onto-> { (/) }  ->  E. x  A  =  { x } )
2322exlimiv 1644 . . 3  |-  ( E. f  f : A -1-1-onto-> { (/)
}  ->  E. x  A  =  { x } )
244, 23sylbi 188 . 2  |-  ( A 
~~  1o  ->  E. x  A  =  { x } )
25 vex 2951 . . . . 5  |-  x  e. 
_V
2625ensn1 7162 . . . 4  |-  { x }  ~~  1o
27 breq1 4207 . . . 4  |-  ( A  =  { x }  ->  ( A  ~~  1o  <->  { x }  ~~  1o ) )
2826, 27mpbiri 225 . . 3  |-  ( A  =  { x }  ->  A  ~~  1o )
2928exlimiv 1644 . 2  |-  ( E. x  A  =  {
x }  ->  A  ~~  1o )
3024, 29impbii 181 1  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725   (/)c0 3620   {csn 3806   <.cop 3809   class class class wbr 4204   `'ccnv 4868   ran crn 4870   -->wf 5441   -onto->wfo 5443   -1-1-onto->wf1o 5444   ` cfv 5445   1oc1o 6708    ~~ cen 7097
This theorem is referenced by:  en1b  7166  reuen1  7167  en2  7335  card1  7844  pm54.43  7876  hash1snb  11669  ufildom1  17946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-1o 6715  df-en 7101
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