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Theorem ficardun 8082
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 7835 . . . . . . 7  |-  ( A  e.  Fin  ->  A  e.  dom  card )
2 finnum 7835 . . . . . . 7  |-  ( B  e.  Fin  ->  B  e.  dom  card )
3 cardacda 8078 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  +c  B )  ~~  (
( card `  A )  +o  ( card `  B
) ) )
41, 2, 3syl2an 464 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  +c  B
)  ~~  ( ( card `  A )  +o  ( card `  B
) ) )
543adant3 977 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  ~~  ( ( card `  A
)  +o  ( card `  B ) ) )
65ensymd 7158 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( (
card `  A )  +o  ( card `  B
) )  ~~  ( A  +c  B ) )
7 cdaun 8052 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
8 entr 7159 . . . 4  |-  ( ( ( ( card `  A
)  +o  ( card `  B ) )  ~~  ( A  +c  B
)  /\  ( A  +c  B )  ~~  ( A  u.  B )
)  ->  ( ( card `  A )  +o  ( card `  B
) )  ~~  ( A  u.  B )
)
96, 7, 8syl2anc 643 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( (
card `  A )  +o  ( card `  B
) )  ~~  ( A  u.  B )
)
10 carden2b 7854 . . 3  |-  ( ( ( card `  A
)  +o  ( card `  B ) )  ~~  ( A  u.  B
)  ->  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( card `  ( A  u.  B )
) )
119, 10syl 16 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( card `  ( A  u.  B )
) )
12 ficardom 7848 . . . 4  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
13 ficardom 7848 . . . 4  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
14 nnacl 6854 . . . . 5  |-  ( ( ( card `  A
)  e.  om  /\  ( card `  B )  e.  om )  ->  (
( card `  A )  +o  ( card `  B
) )  e.  om )
15 cardnn 7850 . . . . 5  |-  ( ( ( card `  A
)  +o  ( card `  B ) )  e. 
om  ->  ( card `  (
( card `  A )  +o  ( card `  B
) ) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
1614, 15syl 16 . . . 4  |-  ( ( ( card `  A
)  e.  om  /\  ( card `  B )  e.  om )  ->  ( card `  ( ( card `  A )  +o  ( card `  B ) ) )  =  ( (
card `  A )  +o  ( card `  B
) ) )
1712, 13, 16syl2an 464 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( card `  (
( card `  A )  +o  ( card `  B
) ) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
18173adant3 977 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( ( card `  A )  +o  ( card `  B ) ) )
1911, 18eqtr3d 2470 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    u. cun 3318    i^i cin 3319   (/)c0 3628   class class class wbr 4212   omcom 4845   dom cdm 4878   ` cfv 5454  (class class class)co 6081    +o coa 6721    ~~ cen 7106   Fincfn 7109   cardccrd 7822    +c ccda 8047
This theorem is referenced by:  hashun  11656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048
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