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Theorem ficardun2 10226
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 5285. (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 10191 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) β‰Ό (𝐴 βŠ” 𝐡))
2 ficardadju 10223 . . . 4 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βŠ” 𝐡) β‰ˆ ((cardβ€˜π΄) +o (cardβ€˜π΅)))
3 domentr 9034 . . . 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 9197 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐡 ∈ Fin) β†’ (𝐴 βˆͺ 𝐡) ∈ Fin)
6 finnum 9972 . . . . 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 9985 . . . . . 6 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
9 ficardom 9985 . . . . . 6 (𝐡 ∈ Fin β†’ (cardβ€˜π΅) ∈ Ο‰)
10 nnacl 8632 . . . . . 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 7876 . . . . 5 (((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ Ο‰ β†’ ((cardβ€˜π΄) +o (cardβ€˜π΅)) ∈ On)
13 onenon 9973 . . . . 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 10001 . . . 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 9987 . . 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 4020 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 1534   ∈ wcel 2099   βˆͺ cun 3945   βŠ† wss 3947   class class class wbr 5148  dom cdm 5678  Oncon0 6369  β€˜cfv 6548  (class class class)co 7420  Ο‰com 7870   +o coa 8484   β‰ˆ cen 8961   β‰Ό cdom 8962  Fincfn 8964   βŠ” cdju 9922  cardccrd 9959
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 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9925  df-card 9963
This theorem is referenced by: (None)
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