Theorem ficardun2 10221

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.) Avoid ax-rep 5254. (Revised by BTernaryTau, 3-Jul-2024.)

Assertion

Ref	Expression
ficardun2	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardun2

Step	Hyp	Ref	Expression
1		undjudom 10187	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
2		ficardadju 10219	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
3		domentr 9032	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
4	1, 2, 3	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
5		unfi 9190	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
6		finnum 9967	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∪ 𝐵) ∈ dom card)
7	5, 6	syl 17	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ dom card)
8		ficardom 9980	. . . . . 6 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
9		ficardom 9980	. . . . . 6 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
10		nnacl 8628	. . . . . 6 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
11	8, 9, 10	syl2an 596	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
12		nnon 7872	. . . . 5 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ ω → ((card‘𝐴) +o (card‘𝐵)) ∈ On)
13		onenon 9968	. . . . 5 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ On → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
14	11, 12, 13	3syl 18	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
15		carddom2 9996	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ∈ dom card ∧ ((card‘𝐴) +o (card‘𝐵)) ∈ dom card) → ((card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
16	7, 14, 15	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
17	4, 16	mpbird 257	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))))
18		cardnn 9982	. . 3 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
19	11, 18	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
20	17, 19	sseqtrd 4000	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3929   ⊆ wss 3931   class class class wbr 5124  dom cdm 5659  Oncon0 6357  ‘cfv 6536  (class class class)co 7410  ωcom 7866   +_o coa 8482   ≈ cen 8961   ≼ cdom 8962  Fincfn 8964   ⊔ cdju 9917  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  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 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9920  df-card 9958

This theorem is referenced by: (None)