Theorem ficardun2 9025

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.)

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		uncdadom 8993	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵))
2		finnum 8774	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
3		finnum 8774	. . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card)
4		cardacda 9020	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
5	2, 3, 4	syl2an 494	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
6		domentr 8015	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 +𝑐 𝐵) ∧ (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵))) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))
7	1, 5, 6	syl2anc 693	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵)))
8		unfi 8227	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
9		finnum 8774	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∪ 𝐵) ∈ dom card)
10	8, 9	syl 17	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ dom card)
11		ficardom 8787	. . . . . 6 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
12		ficardom 8787	. . . . . 6 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
13		nnacl 7691	. . . . . 6 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω)
14	11, 12, 13	syl2an 494	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω)
15		nnon 7071	. . . . 5 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ On)
16		onenon 8775	. . . . 5 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ On → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ dom card)
17	14, 15, 16	3syl 18	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +𝑜 (card‘𝐵)) ∈ dom card)
18		carddom2 8803	. . . 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))))
19	10, 17, 18	syl2anc 693	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))) ↔ (𝐴 ∪ 𝐵) ≼ ((card‘𝐴) +𝑜 (card‘𝐵))))
20	7, 19	mpbird 247	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ (card‘((card‘𝐴) +𝑜 (card‘𝐵))))
21		cardnn 8789	. . 3 ⊢ (((card‘𝐴) +𝑜 (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
22	14, 21	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +𝑜 (card‘𝐵))) = ((card‘𝐴) +𝑜 (card‘𝐵)))
23	20, 22	sseqtrd 3641	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114  Oncon0 5723  ‘cfv 5888  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557   ≈ cen 7952   ≼ cdom 7953  Fincfn 7955  cardccrd 8761   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990

This theorem is referenced by: (None)