MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ficardun2 Unicode version

Theorem ficardun2 8039
Description: The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
Assertion
Ref Expression
ficardun2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( card `  ( A  u.  B )
)  C_  ( ( card `  A )  +o  ( card `  B
) ) )

Proof of Theorem ficardun2
StepHypRef Expression
1 uncdadom 8007 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  ~<_  ( A  +c  B ) )
2 finnum 7791 . . . . 5  |-  ( A  e.  Fin  ->  A  e.  dom  card )
3 finnum 7791 . . . . 5  |-  ( B  e.  Fin  ->  B  e.  dom  card )
4 cardacda 8034 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  +c  B )  ~~  (
( card `  A )  +o  ( card `  B
) ) )
52, 3, 4syl2an 464 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  +c  B
)  ~~  ( ( card `  A )  +o  ( card `  B
) ) )
6 domentr 7125 . . . 4  |-  ( ( ( A  u.  B
)  ~<_  ( A  +c  B )  /\  ( A  +c  B )  ~~  ( ( card `  A
)  +o  ( card `  B ) ) )  ->  ( A  u.  B )  ~<_  ( (
card `  A )  +o  ( card `  B
) ) )
71, 5, 6syl2anc 643 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  ~<_  ( ( card `  A )  +o  ( card `  B ) ) )
8 unfi 7333 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  e.  Fin )
9 finnum 7791 . . . . 5  |-  ( ( A  u.  B )  e.  Fin  ->  ( A  u.  B )  e.  dom  card )
108, 9syl 16 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  e.  dom  card )
11 ficardom 7804 . . . . . 6  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
12 ficardom 7804 . . . . . 6  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
13 nnacl 6813 . . . . . 6  |-  ( ( ( card `  A
)  e.  om  /\  ( card `  B )  e.  om )  ->  (
( card `  A )  +o  ( card `  B
) )  e.  om )
1411, 12, 13syl2an 464 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( card `  A
)  +o  ( card `  B ) )  e. 
om )
15 nnon 4810 . . . . 5  |-  ( ( ( card `  A
)  +o  ( card `  B ) )  e. 
om  ->  ( ( card `  A )  +o  ( card `  B ) )  e.  On )
16 onenon 7792 . . . . 5  |-  ( ( ( card `  A
)  +o  ( card `  B ) )  e.  On  ->  ( ( card `  A )  +o  ( card `  B
) )  e.  dom  card )
1714, 15, 163syl 19 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( card `  A
)  +o  ( card `  B ) )  e. 
dom  card )
18 carddom2 7820 . . . 4  |-  ( ( ( A  u.  B
)  e.  dom  card  /\  ( ( card `  A
)  +o  ( card `  B ) )  e. 
dom  card )  ->  (
( card `  ( A  u.  B ) )  C_  ( card `  ( ( card `  A )  +o  ( card `  B
) ) )  <->  ( A  u.  B )  ~<_  ( (
card `  A )  +o  ( card `  B
) ) ) )
1910, 17, 18syl2anc 643 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( card `  ( A  u.  B )
)  C_  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) )  <-> 
( A  u.  B
)  ~<_  ( ( card `  A )  +o  ( card `  B ) ) ) )
207, 19mpbird 224 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( card `  ( A  u.  B )
)  C_  ( card `  ( ( card `  A
)  +o  ( card `  B ) ) ) )
21 cardnn 7806 . . 3  |-  ( ( ( card `  A
)  +o  ( card `  B ) )  e. 
om  ->  ( card `  (
( card `  A )  +o  ( card `  B
) ) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
2214, 21syl 16 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( card `  (
( card `  A )  +o  ( card `  B
) ) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
2320, 22sseqtrd 3344 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( card `  ( A  u.  B )
)  C_  ( ( card `  A )  +o  ( card `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    u. cun 3278    C_ wss 3280   class class class wbr 4172   Oncon0 4541   omcom 4804   dom cdm 4837   ` cfv 5413  (class class class)co 6040    +o coa 6680    ~~ cen 7065    ~<_ cdom 7066   Fincfn 7068   cardccrd 7778    +c ccda 8003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004
  Copyright terms: Public domain W3C validator