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Theorem ficardun2 8908
 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 8876 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ (𝐴 +𝑐 𝐵))
2 finnum 8657 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
3 finnum 8657 . . . . 5 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
4 cardacda 8903 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
52, 3, 4syl2an 493 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
6 domentr 7901 . . . 4 (((𝐴𝐵) ≼ (𝐴 +𝑐 𝐵) ∧ (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) → (𝐴𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))
71, 5, 6syl2anc 691 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))
8 unfi 8112 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
9 finnum 8657 . . . . 5 ((𝐴𝐵) ∈ Fin → (𝐴𝐵) ∈ dom card)
108, 9syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ dom card)
11 ficardom 8670 . . . . . 6 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
12 ficardom 8670 . . . . . 6 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
13 nnacl 7578 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω)
1411, 12, 13syl2an 493 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω)
15 nnon 6963 . . . . 5 (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ On)
16 onenon 8658 . . . . 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 8686 . . . 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 691 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵))))
207, 19mpbird 246 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))))
21 cardnn 8672 . . 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 3604 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  Oncon0 5640  ‘cfv 5804  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444   ≈ cen 7838   ≼ cdom 7839  Fincfn 7841  cardccrd 8644   +𝑐 ccda 8872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873 This theorem is referenced by: (None)
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