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Theorem ficardunOLD 10195
Description: Obsolete version of ficardun 10194 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 9942 . . . . . . 7 (𝐴 ∈ Fin β†’ 𝐴 ∈ dom card)
2 finnum 9942 . . . . . . 7 (𝐡 ∈ Fin β†’ 𝐡 ∈ dom card)
3 cardadju 10188 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
41, 2, 3syl2an 596 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
543adant3 1132 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
65ensymd 9000 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βŠ” 𝐡))
7 endjudisj 10162 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (𝐴 βŠ” 𝐡) β‰ˆ (𝐴 βˆͺ 𝐡))
8 entr 9001 . . . 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 584 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) β‰ˆ (𝐴 βˆͺ 𝐡))
10 carden2b 9961 . . 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 9955 . . . 4 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
13 ficardom 9955 . . . 4 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
14 nnacl 8610 . . . . 5 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
15 cardnn 9957 . . . . 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 596 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
18173adant3 1132 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
1911, 18eqtr3d 2774 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3946   ∩ cin 3947  βˆ…c0 4322   class class class wbr 5148  dom cdm 5676  β€˜cfv 6543  (class class class)co 7408  Ο‰com 7854   +o coa 8462   β‰ˆ cen 8935  Fincfn 8938   βŠ” cdju 9892  cardccrd 9929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-oadd 8469  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-dju 9895  df-card 9933
This theorem is referenced by: (None)
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