MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ficardun2 Structured version   Visualization version   GIF version

Theorem ficardun2 10194
Description: The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.) Avoid ax-rep 5276. (Revised by BTernaryTau, 3-Jul-2024.)
Assertion
Ref Expression
ficardun2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) βŠ† ((cardβ€˜π΄) +o (cardβ€˜π΅)))

Proof of Theorem ficardun2
StepHypRef Expression
1 undjudom 10159 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) β‰Ό (𝐴 βŠ” 𝐡))
2 ficardadju 10191 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
3 domentr 9006 . . . 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΅)))
41, 2, 3syl2anc 583 . . 3 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) β‰Ό ((cardβ€˜π΄) +o (cardβ€˜π΅)))
5 unfi 9169 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) ∈ Fin)
6 finnum 9940 . . . . 5 ((𝐴 βˆͺ 𝐡) ∈ Fin β†’ (𝐴 βˆͺ 𝐡) ∈ dom card)
75, 6syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) ∈ dom card)
8 ficardom 9953 . . . . . 6 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
9 ficardom 9953 . . . . . 6 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
10 nnacl 8607 . . . . . 6 (((cardβ€˜π΄) ∈ Ο‰ ∧ (cardβ€˜π΅) ∈ Ο‰) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
118, 9, 10syl2an 595 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰)
12 nnon 7855 . . . . 5 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰ β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ On)
13 onenon 9941 . . . . 5 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ On β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ dom card)
1411, 12, 133syl 18 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ dom card)
15 carddom2 9969 . . . 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΅))))
167, 14, 15syl2anc 583 . . 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΅))))
174, 16mpbird 257 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜(𝐴 βˆͺ 𝐡)) βŠ† (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))))
18 cardnn 9955 . . 3 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰ β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
1911, 18syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (cardβ€˜((cardβ€˜π΄) +o (cardβ€˜π΅))) = ((cardβ€˜π΄) +o (cardβ€˜π΅)))
2017, 19sseqtrd 4015 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 395   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3939   βŠ† wss 3941   class class class wbr 5139  dom cdm 5667  Oncon0 6355  β€˜cfv 6534  (class class class)co 7402  Ο‰com 7849   +o coa 8459   β‰ˆ cen 8933   β‰Ό cdom 8934  Fincfn 8936   βŠ” cdju 9890  cardccrd 9927
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-dju 9893  df-card 9931
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator