Theorem ficardun2 10160

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 5229. (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 10126	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵))
2		ficardadju 10158	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
3		domentr 8996	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
4	1, 2, 3	syl2anc 593	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
5		unfi 9141	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
6		finnum 9908	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∪ 𝐵) ∈ dom card)
7	5, 6	syl 17	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ dom card)
8		ficardom 9921	. . . . . 6 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
9		ficardom 9921	. . . . . 6 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
10		nnacl 8583	. . . . . 6 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
11	8, 9, 10	syl2an 605	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
12		nnon 7854	. . . . 5 ⊢ (((card‘𝐴) +o (card‘𝐵)) ∈ ω → ((card‘𝐴) +o (card‘𝐵)) ∈ On)
13		onenon 9909	. . . . 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 9937	. . . 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 593	. . 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 259	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))))
18		cardnn 9923	. . 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 3974	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144   ∪ cun 3904   ⊆ wss 3906   class class class wbr 5102  dom cdm 5649  Oncon0 6348  ‘cfv 6523  (class class class)co 7398  ωcom 7848   +_o coa 8436   ≈ cen 8926   ≼ cdom 8927  Fincfn 8929   ⊔ cdju 9858  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899

This theorem is referenced by: (None)