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Theorem en0 7629
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 7575 . . 3  |-  ( A 
~~  (/)  <->  E. f  f : A -1-1-onto-> (/) )
2 f1ocnv 5824 . . . . 5  |-  ( f : A -1-1-onto-> (/)  ->  `' f : (/)
-1-1-onto-> A )
3 f1o00 5845 . . . . . 6  |-  ( `' f : (/) -1-1-onto-> A  <->  ( `' f  =  (/)  /\  A  =  (/) ) )
43simprbi 466 . . . . 5  |-  ( `' f : (/) -1-1-onto-> A  ->  A  =  (/) )
52, 4syl 17 . . . 4  |-  ( f : A -1-1-onto-> (/)  ->  A  =  (/) )
65exlimiv 1775 . . 3  |-  ( E. f  f : A -1-1-onto-> (/)  ->  A  =  (/) )
71, 6sylbi 199 . 2  |-  ( A 
~~  (/)  ->  A  =  (/) )
8 0ex 4534 . . . 4  |-  (/)  e.  _V
98enref 7599 . . 3  |-  (/)  ~~  (/)
10 breq1 4404 . . 3  |-  ( A  =  (/)  ->  ( A 
~~  (/)  <->  (/)  ~~  (/) ) )
119, 10mpbiri 237 . 2  |-  ( A  =  (/)  ->  A  ~~  (/) )
127, 11impbii 191 1  |-  ( A 
~~  (/)  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1443   E.wex 1662   (/)c0 3730   class class class wbr 4401   `'ccnv 4832   -1-1-onto->wf1o 5580    ~~ cen 7563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-en 7567
This theorem is referenced by:  snfi  7647  dom0  7697  0sdomg  7698  nneneq  7752  snnen2o  7758  enp1i  7803  findcard  7807  findcard2  7808  fiint  7845  cantnff  8176  cantnf0  8177  cantnfp1lem2  8181  cantnflem1  8191  cantnf  8195  cnfcom2lem  8203  cardnueq0  8395  infmap2  8645  fin23lem26  8752  cardeq0  8974  hasheq0  12541  mreexexd  15547  pmtrfmvdn0  17096  pmtrsn  17153  rp-isfinite6  36157
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