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Theorem ficardun2 10179
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 5278. (Revised by BTernaryTau, 3-Jul-2024.)
Assertion
Ref Expression
ficardun2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardun2
StepHypRef Expression
1 undjudom 10144 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ (𝐴𝐵))
2 ficardadju 10176 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
3 domentr 8992 . . . 4 (((𝐴𝐵) ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) → (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
41, 2, 3syl2anc 584 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
5 unfi 9155 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
6 finnum 9925 . . . . 5 ((𝐴𝐵) ∈ Fin → (𝐴𝐵) ∈ dom card)
75, 6syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ dom card)
8 ficardom 9938 . . . . . 6 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
9 ficardom 9938 . . . . . 6 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
10 nnacl 8594 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
118, 9, 10syl2an 596 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
12 nnon 7844 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → ((card‘𝐴) +o (card‘𝐵)) ∈ On)
13 onenon 9926 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ On → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
1411, 12, 133syl 18 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
15 carddom2 9954 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ((card‘𝐴) +o (card‘𝐵)) ∈ dom card) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
167, 14, 15syl2anc 584 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
174, 16mpbird 256 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))))
18 cardnn 9940 . . 3 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
1911, 18syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
2017, 19sseqtrd 4018 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  cun 3942  wss 3944   class class class wbr 5141  dom cdm 5669  Oncon0 6353  cfv 6532  (class class class)co 7393  ωcom 7838   +o coa 8445  cen 8919  cdom 8920  Fincfn 8922  cdju 9875  cardccrd 9912
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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-oadd 8452  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-dju 9878  df-card 9916
This theorem is referenced by: (None)
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