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Theorem en0 6919
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
StepHypRef Expression
1 bren 6866 . . 3  |-  ( A 
~~  (/)  <->  E. f  f : A -1-1-onto-> (/) )
2 f1ocnv 5450 . . . . 5  |-  ( f : A -1-1-onto-> (/)  ->  `' f : (/)
-1-1-onto-> A )
3 f1o00 5473 . . . . . 6  |-  ( `' f : (/) -1-1-onto-> A  <->  ( `' f  =  (/)  /\  A  =  (/) ) )
43simprbi 452 . . . . 5  |-  ( `' f : (/) -1-1-onto-> A  ->  A  =  (/) )
52, 4syl 17 . . . 4  |-  ( f : A -1-1-onto-> (/)  ->  A  =  (/) )
65exlimiv 1670 . . 3  |-  ( E. f  f : A -1-1-onto-> (/)  ->  A  =  (/) )
71, 6sylbi 189 . 2  |-  ( A 
~~  (/)  ->  A  =  (/) )
8 0ex 4151 . . . 4  |-  (/)  e.  _V
98enref 6889 . . 3  |-  (/)  ~~  (/)
10 breq1 4027 . . 3  |-  ( A  =  (/)  ->  ( A 
~~  (/)  <->  (/)  ~~  (/) ) )
119, 10mpbiri 226 . 2  |-  ( A  =  (/)  ->  A  ~~  (/) )
127, 11impbii 182 1  |-  ( A 
~~  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1533    = wceq 1628   (/)c0 3456   class class class wbr 4024   `'ccnv 4687   -1-1-onto->wf1o 5220    ~~ cen 6855
This theorem is referenced by:  snfi  6936  dom0  6984  0sdomg  6985  nneneq  7039  enp1i  7088  findcard  7092  findcard2  7093  fiint  7128  cantnff  7370  cantnf0  7371  cantnfp1lem2  7376  cantnflem1  7386  cantnf  7390  cnfcom2lem  7399  cardnueq0  7592  infmap2  7839  fin23lem26  7946  cardeq0  8169  hasheq0  11347  mreexexd  13544  pmtrfmvdn0  26802
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-en 6859
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