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Theorem hashun3 12420
Description: The size of the union of finite sets is the sum of their sizes minus the size of the intersection. (Contributed by Mario Carneiro, 6-Aug-2017.)
Assertion
Ref Expression
hashun3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B )
)  =  ( ( ( # `  A
)  +  ( # `  B ) )  -  ( # `  ( A  i^i  B ) ) ) )

Proof of Theorem hashun3
StepHypRef Expression
1 diffi 7751 . . . . . . 7  |-  ( B  e.  Fin  ->  ( B  \  A )  e. 
Fin )
21adantl 466 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( B  \  A
)  e.  Fin )
3 simpl 457 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  A  e.  Fin )
4 inss1 3718 . . . . . . 7  |-  ( A  i^i  B )  C_  A
5 ssfi 7740 . . . . . . 7  |-  ( ( A  e.  Fin  /\  ( A  i^i  B ) 
C_  A )  -> 
( A  i^i  B
)  e.  Fin )
63, 4, 5sylancl 662 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  i^i  B
)  e.  Fin )
7 sslin 3724 . . . . . . . . 9  |-  ( ( A  i^i  B ) 
C_  A  ->  (
( B  \  A
)  i^i  ( A  i^i  B ) )  C_  ( ( B  \  A )  i^i  A
) )
84, 7ax-mp 5 . . . . . . . 8  |-  ( ( B  \  A )  i^i  ( A  i^i  B ) )  C_  (
( B  \  A
)  i^i  A )
9 incom 3691 . . . . . . . . 9  |-  ( ( B  \  A )  i^i  A )  =  ( A  i^i  ( B  \  A ) )
10 disjdif 3899 . . . . . . . . 9  |-  ( A  i^i  ( B  \  A ) )  =  (/)
119, 10eqtri 2496 . . . . . . . 8  |-  ( ( B  \  A )  i^i  A )  =  (/)
12 sseq0 3817 . . . . . . . 8  |-  ( ( ( ( B  \  A )  i^i  ( A  i^i  B ) ) 
C_  ( ( B 
\  A )  i^i 
A )  /\  (
( B  \  A
)  i^i  A )  =  (/) )  ->  (
( B  \  A
)  i^i  ( A  i^i  B ) )  =  (/) )
138, 11, 12mp2an 672 . . . . . . 7  |-  ( ( B  \  A )  i^i  ( A  i^i  B ) )  =  (/)
1413a1i 11 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  \  A )  i^i  ( A  i^i  B ) )  =  (/) )
15 hashun 12418 . . . . . 6  |-  ( ( ( B  \  A
)  e.  Fin  /\  ( A  i^i  B )  e.  Fin  /\  (
( B  \  A
)  i^i  ( A  i^i  B ) )  =  (/) )  ->  ( # `  ( ( B  \  A )  u.  ( A  i^i  B ) ) )  =  ( (
# `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) ) )
162, 6, 14, 15syl3anc 1228 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  (
( B  \  A
)  u.  ( A  i^i  B ) ) )  =  ( (
# `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) ) )
17 incom 3691 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
1817uneq2i 3655 . . . . . . . 8  |-  ( ( B  \  A )  u.  ( A  i^i  B ) )  =  ( ( B  \  A
)  u.  ( B  i^i  A ) )
19 uncom 3648 . . . . . . . 8  |-  ( ( B  \  A )  u.  ( B  i^i  A ) )  =  ( ( B  i^i  A
)  u.  ( B 
\  A ) )
20 inundif 3905 . . . . . . . 8  |-  ( ( B  i^i  A )  u.  ( B  \  A ) )  =  B
2118, 19, 203eqtri 2500 . . . . . . 7  |-  ( ( B  \  A )  u.  ( A  i^i  B ) )  =  B
2221a1i 11 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  \  A )  u.  ( A  i^i  B ) )  =  B )
2322fveq2d 5870 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  (
( B  \  A
)  u.  ( A  i^i  B ) ) )  =  ( # `  B ) )
2416, 23eqtr3d 2510 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) )  =  ( # `  B ) )
25 hashcl 12396 . . . . . . 7  |-  ( B  e.  Fin  ->  ( # `
 B )  e. 
NN0 )
2625adantl 466 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  B
)  e.  NN0 )
2726nn0cnd 10854 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  B
)  e.  CC )
28 hashcl 12396 . . . . . . 7  |-  ( ( A  i^i  B )  e.  Fin  ->  ( # `
 ( A  i^i  B ) )  e.  NN0 )
296, 28syl 16 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  i^i  B ) )  e.  NN0 )
3029nn0cnd 10854 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  i^i  B ) )  e.  CC )
31 hashcl 12396 . . . . . . 7  |-  ( ( B  \  A )  e.  Fin  ->  ( # `
 ( B  \  A ) )  e. 
NN0 )
322, 31syl 16 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( B  \  A ) )  e.  NN0 )
3332nn0cnd 10854 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( B  \  A ) )  e.  CC )
3427, 30, 33subadd2d 9949 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( ( # `  B )  -  ( # `
 ( A  i^i  B ) ) )  =  ( # `  ( B  \  A ) )  <-> 
( ( # `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) )  =  ( # `  B ) ) )
3524, 34mpbird 232 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  B
)  -  ( # `  ( A  i^i  B
) ) )  =  ( # `  ( B  \  A ) ) )
3635oveq2d 6300 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  A
)  +  ( (
# `  B )  -  ( # `  ( A  i^i  B ) ) ) )  =  ( ( # `  A
)  +  ( # `  ( B  \  A
) ) ) )
37 hashcl 12396 . . . . 5  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
3837adantr 465 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  A
)  e.  NN0 )
3938nn0cnd 10854 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  A
)  e.  CC )
4039, 27, 30addsubassd 9950 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( ( # `  A )  +  (
# `  B )
)  -  ( # `  ( A  i^i  B
) ) )  =  ( ( # `  A
)  +  ( (
# `  B )  -  ( # `  ( A  i^i  B ) ) ) ) )
41 undif2 3903 . . . 4  |-  ( A  u.  ( B  \  A ) )  =  ( A  u.  B
)
4241fveq2i 5869 . . 3  |-  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( # `  ( A  u.  B
) )
4310a1i 11 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  i^i  ( B  \  A ) )  =  (/) )
44 hashun 12418 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  \  A )  e.  Fin  /\  ( A  i^i  ( B  \  A ) )  =  (/) )  ->  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( (
# `  A )  +  ( # `  ( B  \  A ) ) ) )
453, 2, 43, 44syl3anc 1228 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( ( # `  A )  +  (
# `  ( B  \  A ) ) ) )
4642, 45syl5eqr 2522 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B )
)  =  ( (
# `  A )  +  ( # `  ( B  \  A ) ) ) )
4736, 40, 463eqtr4rd 2519 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B )
)  =  ( ( ( # `  A
)  +  ( # `  B ) )  -  ( # `  ( A  i^i  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ` cfv 5588  (class class class)co 6284   Fincfn 7516    + caddc 9495    - cmin 9805   NN0cn0 10795   #chash 12373
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  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-iun 4327  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-lim 4883  df-suc 4884  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-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-hash 12374
This theorem is referenced by:  incexclem  13611
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