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Theorem cardnueq0 8346
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 8335 . . . 4  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
21ensymd 7567 . . 3  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
3 breq2 4451 . . . 4  |-  ( (
card `  A )  =  (/)  ->  ( A  ~~  ( card `  A
)  <->  A  ~~  (/) ) )
4 en0 7579 . . . 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 5866 . . 3  |-  ( A  =  (/)  ->  ( card `  A )  =  (
card `  (/) ) )
8 card0 8340 . . 3  |-  ( card `  (/) )  =  (/)
97, 8syl6eq 2524 . 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 1379    e. wcel 1767   (/)c0 3785   class class class wbr 4447   dom cdm 4999   ` cfv 5588    ~~ cen 7514   cardccrd 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-er 7312  df-en 7518  df-card 8321
This theorem is referenced by:  carddomi2  8352  cfeq0  8637  cfsuc  8638  sdom2en01  8683
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