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Theorem ficardun2OLD 10224
Description: Obsolete version of ficardun2 10223 as of 3-Jul-2024. (Contributed by Mario Carneiro, 5-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ficardun2OLD ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) βŠ† ((cardβ€˜π΄) +o (cardβ€˜π΅)))

Proof of Theorem ficardun2OLD
StepHypRef Expression
1 undjudom 10188 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) β‰Ό (𝐴 βŠ” 𝐡))
2 finnum 9969 . . . . 5 (𝐴 ∈ Fin β†’ 𝐴 ∈ dom card)
3 finnum 9969 . . . . 5 (𝐡 ∈ Fin β†’ 𝐡 ∈ dom card)
4 cardadju 10215 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
52, 3, 4syl2an 594 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
6 domentr 9030 . . . 4 (((𝐴 βˆͺ 𝐡) β‰Ό (𝐴 βŠ” 𝐡) ∧ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅))) β†’ (𝐴 βˆͺ 𝐡) β‰Ό ((cardβ€˜π΄) +o (cardβ€˜π΅)))
71, 5, 6syl2anc 582 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) β‰Ό ((cardβ€˜π΄) +o (cardβ€˜π΅)))
8 unfi 9193 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) ∈ Fin)
9 finnum 9969 . . . . 5 ((𝐴 βˆͺ 𝐡) ∈ Fin β†’ (𝐴 βˆͺ 𝐡) ∈ dom card)
108, 9syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) ∈ dom card)
11 ficardom 9982 . . . . . 6 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
12 ficardom 9982 . . . . . 6 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
13 nnacl 8628 . . . . . 6 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
1411, 12, 13syl2an 594 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
15 nnon 7872 . . . . 5 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰ β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ On)
16 onenon 9970 . . . . 5 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ On β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ dom card)
1714, 15, 163syl 18 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ dom card)
18 carddom2 9998 . . . 4 (((𝐴 βˆͺ 𝐡) ∈ dom card ∧ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ dom card) β†’ ((cardβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) ↔ (𝐴 βˆͺ 𝐡) β‰Ό ((cardβ€˜π΄) +o (cardβ€˜π΅))))
1910, 17, 18syl2anc 582 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ ((cardβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) ↔ (𝐴 βˆͺ 𝐡) β‰Ό ((cardβ€˜π΄) +o (cardβ€˜π΅))))
207, 19mpbird 256 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))))
21 cardnn 9984 . . 3 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰ β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
2214, 21syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
2320, 22sseqtrd 4012 1 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) βŠ† ((cardβ€˜π΄) +o (cardβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3937   βŠ† wss 3939   class class class wbr 5141  dom cdm 5670  Oncon0 6362  β€˜cfv 6541  (class class class)co 7414  Ο‰com 7866   +o coa 8480   β‰ˆ cen 8957   β‰Ό cdom 8958  Fincfn 8960   βŠ” cdju 9919  cardccrd 9956
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 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-oadd 8487  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-dju 9922  df-card 9960
This theorem is referenced by: (None)
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