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Theorem en0 6924
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 6871 . . 3  |-  ( A 
~~  (/)  <->  E. f  f : A -1-1-onto-> (/) )
2 f1ocnv 5485 . . . . 5  |-  ( f : A -1-1-onto-> (/)  ->  `' f : (/)
-1-1-onto-> A )
3 f1o00 5508 . . . . . 6  |-  ( `' f : (/) -1-1-onto-> A  <->  ( `' f  =  (/)  /\  A  =  (/) ) )
43simprbi 450 . . . . 5  |-  ( `' f : (/) -1-1-onto-> A  ->  A  =  (/) )
52, 4syl 15 . . . 4  |-  ( f : A -1-1-onto-> (/)  ->  A  =  (/) )
65exlimiv 1666 . . 3  |-  ( E. f  f : A -1-1-onto-> (/)  ->  A  =  (/) )
71, 6sylbi 187 . 2  |-  ( A 
~~  (/)  ->  A  =  (/) )
8 0ex 4150 . . . 4  |-  (/)  e.  _V
98enref 6894 . . 3  |-  (/)  ~~  (/)
10 breq1 4026 . . 3  |-  ( A  =  (/)  ->  ( A 
~~  (/)  <->  (/)  ~~  (/) ) )
119, 10mpbiri 224 . 2  |-  ( A  =  (/)  ->  A  ~~  (/) )
127, 11impbii 180 1  |-  ( A 
~~  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    = wceq 1623   (/)c0 3455   class class class wbr 4023   `'ccnv 4688   -1-1-onto->wf1o 5254    ~~ cen 6860
This theorem is referenced by:  snfi  6941  dom0  6989  0sdomg  6990  nneneq  7044  enp1i  7093  findcard  7097  findcard2  7098  fiint  7133  cantnff  7375  cantnf0  7376  cantnfp1lem2  7381  cantnflem1  7391  cantnf  7395  cnfcom2lem  7404  cardnueq0  7597  infmap2  7844  fin23lem26  7951  cardeq0  8174  hasheq0  11353  mreexexd  13550  pmtrfmvdn0  27403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864
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