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Theorem ficardun 10192
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 5276. (Revised by BTernaryTau, 3-Jul-2024.)
Assertion
Ref Expression
ficardun ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))

Proof of Theorem ficardun
StepHypRef Expression
1 ficardadju 10191 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
213adant3 1129 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
32ensymd 8998 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βŠ” 𝐡))
4 endjudisj 10160 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ 𝐡))
5 entr 8999 . . . 4 ((((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βŠ” 𝐡) ∧ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ 𝐡)) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βˆͺ 𝐡))
63, 4, 5syl2anc 583 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βˆͺ 𝐡))
7 carden2b 9959 . . 3 (((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βˆͺ 𝐡) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = (cardβ€˜(𝐴 βˆͺ 𝐡)))
86, 7syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = (cardβ€˜(𝐴 βˆͺ 𝐡)))
9 ficardom 9953 . . . 4 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
10 ficardom 9953 . . . 4 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
11 nnacl 8607 . . . . 5 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
12 cardnn 9955 . . . . 5 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰ β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
1311, 12syl 17 . . . 4 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
149, 10, 13syl2an 595 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
15143adant3 1129 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
168, 15eqtr3d 2766 1 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3939   ∩ cin 3940  βˆ…c0 4315   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  Ο‰com 7849   +o coa 8459   β‰ˆ cen 8933  Fincfn 8936   βŠ” cdju 9890  cardccrd 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-dju 9893  df-card 9931
This theorem is referenced by:  hashun  14343
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