Theorem ficardun 10248

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.) Avoid ax-rep 5288. (Revised by BTernaryTau, 3-Jul-2024.)

Assertion

Ref	Expression
ficardun	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardun

Step	Hyp	Ref	Expression
1		ficardadju 10247	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
2	1	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
3	2	ensymd 9053	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
4		endjudisj 10216	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))
5		entr 9054	. . . 4 ⊢ ((((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵)) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ∪ 𝐵))
6	3, 4, 5	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ∪ 𝐵))
7		carden2b 10014	. . 3 ⊢ (((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ∪ 𝐵) → (card‘((card‘𝐴) +o (card‘𝐵))) = (card‘(𝐴 ∪ 𝐵)))
8	6, 7	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘𝐴) +o (card‘𝐵))) = (card‘(𝐴 ∪ 𝐵)))
9		ficardom 10008	. . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
10		ficardom 10008	. . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
11		nnacl 8657	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
12		cardnn 10010	. . . . 5 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
13	11, 12	syl 17	. . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
14	9, 10, 13	syl2an 596	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
15	14	3adant3 1133	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
16	8, 15	eqtr3d 2779	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +o (card‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108   ∪ cun 3964   ∩ cin 3965  ∅c0 4342   class class class wbr 5151  ‘cfv 6569  (class class class)co 7438  ωcom 7894   +_o coa 8511   ≈ cen 8990  Fincfn 8993   ⊔ cdju 9945  cardccrd 9982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-tr 5269  df-id 5587  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-we 5647  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-ord 6395  df-on 6396  df-lim 6397  df-suc 6398  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-om 7895  df-1st 8022  df-2nd 8023  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-rdg 8458  df-1o 8514  df-oadd 8518  df-er 8753  df-en 8994  df-dom 8995  df-sdom 8996  df-fin 8997  df-dju 9948  df-card 9986

This theorem is referenced by:  hashun  14427