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Theorem ficardun2OLD 9817
Description: Obsolete version of ficardun2 9816 as of 3-Jul-2024. (Contributed by Mario Carneiro, 5-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ficardun2OLD ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardun2OLD
StepHypRef Expression
1 undjudom 9781 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ (𝐴𝐵))
2 finnum 9564 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
3 finnum 9564 . . . . 5 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
4 cardadju 9808 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
52, 3, 4syl2an 599 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
6 domentr 8687 . . . 4 (((𝐴𝐵) ≼ (𝐴𝐵) ∧ (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) → (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
71, 5, 6syl2anc 587 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵)))
8 unfi 8850 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
9 finnum 9564 . . . . 5 ((𝐴𝐵) ∈ Fin → (𝐴𝐵) ∈ dom card)
108, 9syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ dom card)
11 ficardom 9577 . . . . . 6 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
12 ficardom 9577 . . . . . 6 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
13 nnacl 8339 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
1411, 12, 13syl2an 599 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
15 nnon 7650 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → ((card‘𝐴) +o (card‘𝐵)) ∈ On)
16 onenon 9565 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ On → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
1714, 15, 163syl 18 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ dom card)
18 carddom2 9593 . . . 4 (((𝐴𝐵) ∈ dom card ∧ ((card‘𝐴) +o (card‘𝐵)) ∈ dom card) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
1910, 17, 18syl2anc 587 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))) ↔ (𝐴𝐵) ≼ ((card‘𝐴) +o (card‘𝐵))))
207, 19mpbird 260 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ (card‘((card‘𝐴) +o (card‘𝐵))))
21 cardnn 9579 . . 3 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
2214, 21syl 17 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘((card‘𝐴) +o (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
2320, 22sseqtrd 3941 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  cun 3864  wss 3866   class class class wbr 5053  dom cdm 5551  Oncon0 6213  cfv 6380  (class class class)co 7213  ωcom 7644   +o coa 8199  cen 8623  cdom 8624  Fincfn 8626  cdju 9514  cardccrd 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-dju 9517  df-card 9555
This theorem is referenced by: (None)
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