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Theorem ficardun2 9313
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.)
Assertion
Ref Expression
ficardun2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵)))

Proof of Theorem ficardun2
StepHypRef Expression
1 uncdadom 9281 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))
2 finnum 9060 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
3 finnum 9060 . . . . 5 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
4 cardacda 9308 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
52, 3, 4syl2an 590 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
6 domentr 8254 . . . 4 (((𝐴𝐵) ≼ (𝐴 +𝑐 𝐵) ∧ (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) → (𝐴𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))
71, 5, 6syl2anc 580 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))
8 unfi 8469 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
9 finnum 9060 . . . . 5 ((𝐴𝐵) ∈ Fin → (𝐴𝐵) ∈ dom card)
108, 9syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ dom card)
11 ficardom 9073 . . . . . 6 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
12 ficardom 9073 . . . . . 6 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
13 nnacl 7931 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω)
1411, 12, 13syl2an 590 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω)
15 nnon 7305 . . . . 5 (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ On)
16 onenon 9061 . . . . 5 (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ On → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ dom card)
1714, 15, 163syl 18 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ dom card)
18 carddom2 9089 . . . 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))))
1910, 17, 18syl2anc 580 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵))))
207, 19mpbird 249 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))))
21 cardnn 9075 . . 3 (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
2214, 21syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
2320, 22sseqtrd 3837 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cun 3767  wss 3769   class class class wbr 4843  dom cdm 5312  Oncon0 5941  cfv 6101  (class class class)co 6878  ωcom 7299   +𝑜 coa 7796  cen 8192  cdom 8193  Fincfn 8195  cardccrd 9047   +𝑐 ccda 9277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-cda 9278
This theorem is referenced by: (None)
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