MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ficardun Structured version   Visualization version   GIF version

Theorem ficardun 10243
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.) Avoid ax-rep 5290. (Revised by BTernaryTau, 3-Jul-2024.)
Assertion
Ref Expression
ficardun ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘(𝐴𝐵)) = ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardun
StepHypRef Expression
1 ficardadju 10242 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
213adant3 1129 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
32ensymd 9036 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
4 endjudisj 10211 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (𝐴𝐵))
5 entr 9037 . . . 4 ((((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵) ∧ (𝐴𝐵) ≈ (𝐴𝐵)) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
63, 4, 5syl2anc 582 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
7 carden2b 10010 . . 3 (((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵) → (card‘((card‘𝐴) +o (card‘𝐵))) = (card‘(𝐴𝐵)))
86, 7syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘((card‘𝐴) +o (card‘𝐵))) = (card‘(𝐴𝐵)))
9 ficardom 10004 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
10 ficardom 10004 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
11 nnacl 8641 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
12 cardnn 10006 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
1311, 12syl 17 . . . 4 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
149, 10, 13syl2an 594 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
15143adant3 1129 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
168, 15eqtr3d 2768 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘(𝐴𝐵)) = ((card‘𝐴) +o (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  cun 3945  cin 3946  c0 4325   class class class wbr 5153  cfv 6554  (class class class)co 7424  ωcom 7876   +o coa 8493  cen 8971  Fincfn 8974  cdju 9941  cardccrd 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-oadd 8500  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-dju 9944  df-card 9982
This theorem is referenced by:  hashun  14399
  Copyright terms: Public domain W3C validator