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Theorem en0 7129
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 7076 . . 3  |-  ( A 
~~  (/)  <->  E. f  f : A -1-1-onto-> (/) )
2 f1ocnv 5646 . . . . 5  |-  ( f : A -1-1-onto-> (/)  ->  `' f : (/)
-1-1-onto-> A )
3 f1o00 5669 . . . . . 6  |-  ( `' f : (/) -1-1-onto-> A  <->  ( `' f  =  (/)  /\  A  =  (/) ) )
43simprbi 451 . . . . 5  |-  ( `' f : (/) -1-1-onto-> A  ->  A  =  (/) )
52, 4syl 16 . . . 4  |-  ( f : A -1-1-onto-> (/)  ->  A  =  (/) )
65exlimiv 1641 . . 3  |-  ( E. f  f : A -1-1-onto-> (/)  ->  A  =  (/) )
71, 6sylbi 188 . 2  |-  ( A 
~~  (/)  ->  A  =  (/) )
8 0ex 4299 . . . 4  |-  (/)  e.  _V
98enref 7099 . . 3  |-  (/)  ~~  (/)
10 breq1 4175 . . 3  |-  ( A  =  (/)  ->  ( A 
~~  (/)  <->  (/)  ~~  (/) ) )
119, 10mpbiri 225 . 2  |-  ( A  =  (/)  ->  A  ~~  (/) )
127, 11impbii 181 1  |-  ( A 
~~  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1547    = wceq 1649   (/)c0 3588   class class class wbr 4172   `'ccnv 4836   -1-1-onto->wf1o 5412    ~~ cen 7065
This theorem is referenced by:  snfi  7146  dom0  7194  0sdomg  7195  nneneq  7249  enp1i  7302  findcard  7306  findcard2  7307  fiint  7342  cantnff  7585  cantnf0  7586  cantnfp1lem2  7591  cantnflem1  7601  cantnf  7605  cnfcom2lem  7614  cardnueq0  7807  infmap2  8054  fin23lem26  8161  cardeq0  8383  hasheq0  11599  mreexexd  13828  pmtrfmvdn0  27271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-en 7069
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