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Theorem cardnueq0 8336
Description: The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardnueq0  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )

Proof of Theorem cardnueq0
StepHypRef Expression
1 cardid2 8325 . . . 4  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
21ensymd 7559 . . 3  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
3 breq2 4443 . . . 4  |-  ( (
card `  A )  =  (/)  ->  ( A  ~~  ( card `  A
)  <->  A  ~~  (/) ) )
4 en0 7571 . . . 4  |-  ( A 
~~  (/)  <->  A  =  (/) )
53, 4syl6bb 261 . . 3  |-  ( (
card `  A )  =  (/)  ->  ( A  ~~  ( card `  A
)  <->  A  =  (/) ) )
62, 5syl5ibcom 220 . 2  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  ->  A  =  (/) ) )
7 fveq2 5848 . . 3  |-  ( A  =  (/)  ->  ( card `  A )  =  (
card `  (/) ) )
8 card0 8330 . . 3  |-  ( card `  (/) )  =  (/)
97, 8syl6eq 2511 . 2  |-  ( A  =  (/)  ->  ( card `  A )  =  (/) )
106, 9impbid1 203 1  |-  ( A  e.  dom  card  ->  ( ( card `  A
)  =  (/)  <->  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   (/)c0 3783   class class class wbr 4439   dom cdm 4988   ` cfv 5570    ~~ cen 7506   cardccrd 8307
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-3or 972  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-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  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-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-er 7303  df-en 7510  df-card 8311
This theorem is referenced by:  carddomi2  8342  cfeq0  8627  cfsuc  8628  sdom2en01  8673
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