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Theorem ficardun 10144
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 5246. (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 10143 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
213adant3 1133 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
32ensymd 8951 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βŠ” 𝐡))
4 endjudisj 10112 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ 𝐡))
5 entr 8952 . . . 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 585 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βˆͺ 𝐡))
7 carden2b 9911 . . 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 9905 . . . 4 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
10 ficardom 9905 . . . 4 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
11 nnacl 8562 . . . . 5 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
12 cardnn 9907 . . . . 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 597 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
15143adant3 1133 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
168, 15eqtr3d 2775 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3912   ∩ cin 3913  βˆ…c0 4286   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  Ο‰com 7806   +o coa 8413   β‰ˆ cen 8886  Fincfn 8889   βŠ” cdju 9842  cardccrd 9879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-oadd 8420  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-dju 9845  df-card 9883
This theorem is referenced by:  hashun  14291
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