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Theorem hashun3 12438
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 7744 . . . . . . 7  |-  ( B  e.  Fin  ->  ( B  \  A )  e. 
Fin )
21adantl 464 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( B  \  A
)  e.  Fin )
3 simpl 455 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  A  e.  Fin )
4 inss1 3704 . . . . . . 7  |-  ( A  i^i  B )  C_  A
5 ssfi 7733 . . . . . . 7  |-  ( ( A  e.  Fin  /\  ( A  i^i  B ) 
C_  A )  -> 
( A  i^i  B
)  e.  Fin )
63, 4, 5sylancl 660 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  i^i  B
)  e.  Fin )
7 sslin 3710 . . . . . . . . 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 3677 . . . . . . . . 9  |-  ( ( B  \  A )  i^i  A )  =  ( A  i^i  ( B  \  A ) )
10 disjdif 3888 . . . . . . . . 9  |-  ( A  i^i  ( B  \  A ) )  =  (/)
119, 10eqtri 2483 . . . . . . . 8  |-  ( ( B  \  A )  i^i  A )  =  (/)
12 sseq0 3816 . . . . . . . 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 670 . . . . . . 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 12436 . . . . . 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 1226 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  (
( B  \  A
)  u.  ( A  i^i  B ) ) )  =  ( (
# `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) ) )
17 incom 3677 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
1817uneq2i 3641 . . . . . . . 8  |-  ( ( B  \  A )  u.  ( A  i^i  B ) )  =  ( ( B  \  A
)  u.  ( B  i^i  A ) )
19 uncom 3634 . . . . . . . 8  |-  ( ( B  \  A )  u.  ( B  i^i  A ) )  =  ( ( B  i^i  A
)  u.  ( B 
\  A ) )
20 inundif 3894 . . . . . . . 8  |-  ( ( B  i^i  A )  u.  ( B  \  A ) )  =  B
2118, 19, 203eqtri 2487 . . . . . . 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 5852 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  (
( B  \  A
)  u.  ( A  i^i  B ) ) )  =  ( # `  B ) )
2416, 23eqtr3d 2497 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) )  =  ( # `  B ) )
25 hashcl 12413 . . . . . . 7  |-  ( B  e.  Fin  ->  ( # `
 B )  e. 
NN0 )
2625adantl 464 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  B
)  e.  NN0 )
2726nn0cnd 10850 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  B
)  e.  CC )
28 hashcl 12413 . . . . . . 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 10850 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  i^i  B ) )  e.  CC )
31 hashcl 12413 . . . . . . 7  |-  ( ( B  \  A )  e.  Fin  ->  ( # `
 ( B  \  A ) )  e. 
NN0 )
322, 31syl 16 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( B  \  A ) )  e.  NN0 )
3332nn0cnd 10850 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( B  \  A ) )  e.  CC )
3427, 30, 33subadd2d 9941 . . . 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 6286 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  A
)  +  ( (
# `  B )  -  ( # `  ( A  i^i  B ) ) ) )  =  ( ( # `  A
)  +  ( # `  ( B  \  A
) ) ) )
37 hashcl 12413 . . . . 5  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
3837adantr 463 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  A
)  e.  NN0 )
3938nn0cnd 10850 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  A
)  e.  CC )
4039, 27, 30addsubassd 9942 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( ( # `  A )  +  (
# `  B )
)  -  ( # `  ( A  i^i  B
) ) )  =  ( ( # `  A
)  +  ( (
# `  B )  -  ( # `  ( A  i^i  B ) ) ) ) )
41 undif2 3892 . . . 4  |-  ( A  u.  ( B  \  A ) )  =  ( A  u.  B
)
4241fveq2i 5851 . . 3  |-  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( # `  ( A  u.  B
) )
4310a1i 11 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  i^i  ( B  \  A ) )  =  (/) )
44 hashun 12436 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  \  A )  e.  Fin  /\  ( A  i^i  ( B  \  A ) )  =  (/) )  ->  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( (
# `  A )  +  ( # `  ( B  \  A ) ) ) )
453, 2, 43, 44syl3anc 1226 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( ( # `  A )  +  (
# `  ( B  \  A ) ) ) )
4642, 45syl5eqr 2509 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B )
)  =  ( (
# `  A )  +  ( # `  ( B  \  A ) ) ) )
4736, 40, 463eqtr4rd 2506 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 367    = wceq 1398    e. wcel 1823    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   ` cfv 5570  (class class class)co 6270   Fincfn 7509    + caddc 9484    - cmin 9796   NN0cn0 10791   #chash 12390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-hash 12391
This theorem is referenced by:  incexclem  13733
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