Theorem ficardun 10223

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 5259. (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 10222	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
2	1	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
3	2	ensymd 9027	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
4		endjudisj 10191	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))
5		entr 9028	. . . 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 9989	. . 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 9983	. . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
10		ficardom 9983	. . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
11		nnacl 8631	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
12		cardnn 9985	. . . . 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 1132	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
16	8, 15	eqtr3d 2771	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +o (card‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ∪ cun 3929   ∩ cin 3930  ∅c0 4313   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413  ωcom 7869   +_o coa 8485   ≈ cen 8964  Fincfn 8967   ⊔ cdju 9920  cardccrd 9957

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-oadd 8492  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-dju 9923  df-card 9961

This theorem is referenced by:  hashun  14403