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Theorem en0 7571
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 7518 . . 3  |-  ( A 
~~  (/)  <->  E. f  f : A -1-1-onto-> (/) )
2 f1ocnv 5810 . . . . 5  |-  ( f : A -1-1-onto-> (/)  ->  `' f : (/)
-1-1-onto-> A )
3 f1o00 5830 . . . . . 6  |-  ( `' f : (/) -1-1-onto-> A  <->  ( `' f  =  (/)  /\  A  =  (/) ) )
43simprbi 462 . . . . 5  |-  ( `' f : (/) -1-1-onto-> A  ->  A  =  (/) )
52, 4syl 16 . . . 4  |-  ( f : A -1-1-onto-> (/)  ->  A  =  (/) )
65exlimiv 1727 . . 3  |-  ( E. f  f : A -1-1-onto-> (/)  ->  A  =  (/) )
71, 6sylbi 195 . 2  |-  ( A 
~~  (/)  ->  A  =  (/) )
8 0ex 4569 . . . 4  |-  (/)  e.  _V
98enref 7541 . . 3  |-  (/)  ~~  (/)
10 breq1 4442 . . 3  |-  ( A  =  (/)  ->  ( A 
~~  (/)  <->  (/)  ~~  (/) ) )
119, 10mpbiri 233 . 2  |-  ( A  =  (/)  ->  A  ~~  (/) )
127, 11impbii 188 1  |-  ( A 
~~  (/)  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398   E.wex 1617   (/)c0 3783   class class class wbr 4439   `'ccnv 4987   -1-1-onto->wf1o 5569    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-en 7510
This theorem is referenced by:  snfi  7589  dom0  7638  0sdomg  7639  nneneq  7693  snnen2o  7699  enp1i  7747  findcard  7751  findcard2  7752  fiint  7789  cantnff  8084  cantnf0  8085  cantnfp1lem2  8089  cantnflem1  8099  cantnf  8103  cantnfp1lem2OLD  8115  cantnflem1OLD  8122  cantnfOLD  8125  cnfcom2lem  8136  cnfcom2lemOLD  8144  cardnueq0  8336  infmap2  8589  fin23lem26  8696  cardeq0  8918  hasheq0  12419  mreexexd  15140  pmtrfmvdn0  16689  pmtrsn  16746  rp-isfinite6  38176
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