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Theorem ficardun 8907
 Description: The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ficardun ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘(𝐴𝐵)) = ((card‘𝐴) +𝑜 (card‘𝐵)))

Proof of Theorem ficardun
StepHypRef Expression
1 finnum 8657 . . . . . . 7 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
2 finnum 8657 . . . . . . 7 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
3 cardacda 8903 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
41, 2, 3syl2an 493 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
543adant3 1074 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
65ensymd 7893 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵))
7 cdaun 8877 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))
8 entr 7894 . . . 4 ((((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴 +𝑐 𝐵) ∧ (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵)) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴𝐵))
96, 7, 8syl2anc 691 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → ((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴𝐵))
10 carden2b 8676 . . 3 (((card‘𝐴) +𝑜 (card‘𝐵)) ≈ (𝐴𝐵) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = (card‘(𝐴𝐵)))
119, 10syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = (card‘(𝐴𝐵)))
12 ficardom 8670 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
13 ficardom 8670 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
14 nnacl 7578 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω)
15 cardnn 8672 . . . . 5 (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
1614, 15syl 17 . . . 4 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
1712, 13, 16syl2an 493 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
18173adant3 1074 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
1911, 18eqtr3d 2646 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘(𝐴𝐵)) = ((card‘𝐴) +𝑜 (card‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ∩ cin 3539  ∅c0 3874   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444   ≈ cen 7838  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:  hashun  13032
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