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Theorem en0 6682
Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
en0  |-  ( A 
~~  (/)  <->  A  =  (/) )

Proof of Theorem en0
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 bren 6634 . . 3  |-  ( A 
~~  (/)  <->  E. f  f : A -1-1-onto-> (/) )
2 f1ocnv 5373 . . . . 5  |-  ( f : A -1-1-onto-> (/)  ->  `' f : (/)
-1-1-onto-> A )
3 f1o00 5395 . . . . . 6  |-  ( `' f : (/) -1-1-onto-> A  <->  ( `' f  =  (/)  /\  A  =  (/) ) )
43simprbi 273 . . . . 5  |-  ( `' f : (/) -1-1-onto-> A  ->  A  =  (/) )
52, 4syl 14 . . . 4  |-  ( f : A -1-1-onto-> (/)  ->  A  =  (/) )
65exlimiv 1577 . . 3  |-  ( E. f  f : A -1-1-onto-> (/)  ->  A  =  (/) )
71, 6sylbi 120 . 2  |-  ( A 
~~  (/)  ->  A  =  (/) )
8 0ex 4050 . . . 4  |-  (/)  e.  _V
98enref 6652 . . 3  |-  (/)  ~~  (/)
10 breq1 3927 . . 3  |-  ( A  =  (/)  ->  ( A 
~~  (/)  <->  (/)  ~~  (/) ) )
119, 10mpbiri 167 . 2  |-  ( A  =  (/)  ->  A  ~~  (/) )
127, 11impbii 125 1  |-  ( A 
~~  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331   E.wex 1468   (/)c0 3358   class class class wbr 3924   `'ccnv 4533   -1-1-onto->wf1o 5117    ~~ cen 6625
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-en 6628
This theorem is referenced by:  nneneq  6744  php5  6745  snnen2oprc  6747  php5dom  6750  ssfilem  6762  dif1enen  6767  fin0  6772  fin0or  6773  diffitest  6774  findcard  6775  findcard2  6776  findcard2s  6777  diffisn  6780  fiintim  6810  fisseneq  6813  fihasheq0  10533  zfz1iso  10577
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