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Theorem ficardun 7844
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 7597 . . . . . . 7  |-  ( A  e.  Fin  ->  A  e.  dom  card )
2 finnum 7597 . . . . . . 7  |-  ( B  e.  Fin  ->  B  e.  dom  card )
3 cardacda 7840 . . . . . . 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 6926 . . . . 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 7814 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
9 entr 6929 . . . 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 7616 . . 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 7610 . . . 4  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
14 ficardom 7610 . . . 4  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
15 nnacl 6625 . . . . 5  |-  ( ( ( card `  A
)  e.  om  /\  ( card `  B )  e.  om )  ->  (
( card `  A )  +o  ( card `  B
) )  e.  om )
16 cardnn 7612 . . . . 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 2330 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 1632    e. wcel 1696    u. cun 3163    i^i cin 3164   (/)c0 3468   class class class wbr 4039   omcom 4672   dom cdm 4705   ` cfv 5271  (class class class)co 5874    +o coa 6492    ~~ cen 6876   Fincfn 6879   cardccrd 7584    +c ccda 7809
This theorem is referenced by:  hashun  11380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810
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