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Theorem en0 7012
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 6959 . . 3  |-  ( A 
~~  (/)  <->  E. f  f : A -1-1-onto-> (/) )
2 f1ocnv 5568 . . . . 5  |-  ( f : A -1-1-onto-> (/)  ->  `' f : (/)
-1-1-onto-> A )
3 f1o00 5591 . . . . . 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 1634 . . 3  |-  ( E. f  f : A -1-1-onto-> (/)  ->  A  =  (/) )
71, 6sylbi 187 . 2  |-  ( A 
~~  (/)  ->  A  =  (/) )
8 0ex 4231 . . . 4  |-  (/)  e.  _V
98enref 6982 . . 3  |-  (/)  ~~  (/)
10 breq1 4107 . . 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 1541    = wceq 1642   (/)c0 3531   class class class wbr 4104   `'ccnv 4770   -1-1-onto->wf1o 5336    ~~ cen 6948
This theorem is referenced by:  snfi  7029  dom0  7077  0sdomg  7078  nneneq  7132  enp1i  7183  findcard  7187  findcard2  7188  fiint  7223  cantnff  7465  cantnf0  7466  cantnfp1lem2  7471  cantnflem1  7481  cantnf  7485  cnfcom2lem  7494  cardnueq0  7687  infmap2  7934  fin23lem26  8041  cardeq0  8264  hasheq0  11446  mreexexd  13649  pmtrfmvdn0  26726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-en 6952
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