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Theorem ficardun 10223
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 5285. (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 10222 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
213adant3 1130 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
32ensymd 9025 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βŠ” 𝐡))
4 endjudisj 10191 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ 𝐡))
5 entr 9026 . . . 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 9990 . . 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 9984 . . . 4 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
10 ficardom 9984 . . . 4 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
11 nnacl 8631 . . . . 5 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
12 cardnn 9986 . . . . 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 1130 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
168, 15eqtr3d 2770 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 1085   = wceq 1534   ∈ wcel 2099   βˆͺ cun 3945   ∩ cin 3946  βˆ…c0 4323   class class class wbr 5148  β€˜cfv 6548  (class class class)co 7420  Ο‰com 7870   +o coa 8483   β‰ˆ cen 8960  Fincfn 8963   βŠ” cdju 9921  cardccrd 9958
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 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-oadd 8490  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-dju 9924  df-card 9962
This theorem is referenced by:  hashun  14373
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