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Theorem ficardunOLD 10198
Description: Obsolete version of ficardun 10197 as of 3-Jul-2024. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ficardunOLD ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))

Proof of Theorem ficardunOLD
StepHypRef Expression
1 finnum 9945 . . . . . . 7 (𝐴 ∈ Fin β†’ 𝐴 ∈ dom card)
2 finnum 9945 . . . . . . 7 (𝐡 ∈ Fin β†’ 𝐡 ∈ dom card)
3 cardadju 10191 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
41, 2, 3syl2an 595 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
543adant3 1129 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
65ensymd 9003 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βŠ” 𝐡))
7 endjudisj 10165 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ 𝐡))
8 entr 9004 . . . 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))
96, 7, 8syl2anc 583 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βˆͺ 𝐡))
10 carden2b 9964 . . 3 (((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βˆͺ 𝐡) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = (cardβ€˜(𝐴 βˆͺ 𝐡)))
119, 10syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = (cardβ€˜(𝐴 βˆͺ 𝐡)))
12 ficardom 9958 . . . 4 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
13 ficardom 9958 . . . 4 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
14 nnacl 8612 . . . . 5 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
15 cardnn 9960 . . . . 5 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰ β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
1614, 15syl 17 . . . 4 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
1712, 13, 16syl2an 595 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
18173adant3 1129 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
1911, 18eqtr3d 2768 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 3941   ∩ cin 3942  βˆ…c0 4317   class class class wbr 5141  dom cdm 5669  β€˜cfv 6537  (class class class)co 7405  Ο‰com 7852   +o coa 8464   β‰ˆ cen 8938  Fincfn 8941   βŠ” cdju 9895  cardccrd 9932
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936
This theorem is referenced by: (None)
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