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Theorem ficardun 7828
Description: The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ficardun  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( A  u.  B
) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )

Proof of Theorem ficardun
StepHypRef Expression
1 finnum 7581 . . . . . . 7  |-  ( A  e.  Fin  ->  A  e.  dom  card )
2 finnum 7581 . . . . . . 7  |-  ( B  e.  Fin  ->  B  e.  dom  card )
3 cardacda 7824 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  +c  B )  ~~  (
( card `  A )  +o  ( card `  B
) ) )
41, 2, 3syl2an 463 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  +c  B
)  ~~  ( ( card `  A )  +o  ( card `  B
) ) )
543adant3 975 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  ~~  ( ( card `  A
)  +o  ( card `  B ) ) )
6 ensym 6910 . . . . 5  |-  ( ( A  +c  B ) 
~~  ( ( card `  A )  +o  ( card `  B ) )  ->  ( ( card `  A )  +o  ( card `  B ) ) 
~~  ( A  +c  B ) )
75, 6syl 15 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( (
card `  A )  +o  ( card `  B
) )  ~~  ( A  +c  B ) )
8 cdaun 7798 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
9 entr 6913 . . . 4  |-  ( ( ( ( card `  A
)  +o  ( card `  B ) )  ~~  ( A  +c  B
)  /\  ( A  +c  B )  ~~  ( A  u.  B )
)  ->  ( ( card `  A )  +o  ( card `  B
) )  ~~  ( A  u.  B )
)
107, 8, 9syl2anc 642 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( (
card `  A )  +o  ( card `  B
) )  ~~  ( A  u.  B )
)
11 carden2b 7600 . . 3  |-  ( ( ( card `  A
)  +o  ( card `  B ) )  ~~  ( A  u.  B
)  ->  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( card `  ( A  u.  B )
) )
1210, 11syl 15 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( card `  ( A  u.  B )
) )
13 ficardom 7594 . . . 4  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
14 ficardom 7594 . . . 4  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
15 nnacl 6609 . . . . 5  |-  ( ( ( card `  A
)  e.  om  /\  ( card `  B )  e.  om )  ->  (
( card `  A )  +o  ( card `  B
) )  e.  om )
16 cardnn 7596 . . . . 5  |-  ( ( ( card `  A
)  +o  ( card `  B ) )  e. 
om  ->  ( card `  (
( card `  A )  +o  ( card `  B
) ) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
1715, 16syl 15 . . . 4  |-  ( ( ( card `  A
)  e.  om  /\  ( card `  B )  e.  om )  ->  ( card `  ( ( card `  A )  +o  ( card `  B ) ) )  =  ( (
card `  A )  +o  ( card `  B
) ) )
1813, 14, 17syl2an 463 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( card `  (
( card `  A )  +o  ( card `  B
) ) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
19183adant3 975 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( ( card `  A )  +o  ( card `  B ) ) )
2012, 19eqtr3d 2317 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( A  u.  B
) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150    i^i cin 3151   (/)c0 3455   class class class wbr 4023   omcom 4656   dom cdm 4689   ` cfv 5255  (class class class)co 5858    +o coa 6476    ~~ cen 6860   Fincfn 6863   cardccrd 7568    +c ccda 7793
This theorem is referenced by:  hashun  11364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794
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