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Theorem ficardun2OLD 9621
 Description: Obsolete version of ficardun2 9620 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 9585 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≼ (𝐴𝐵))
2 finnum 9368 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
3 finnum 9368 . . . . 5 (𝐵 ∈ Fin → 𝐵 ∈ dom card)
4 cardadju 9612 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
52, 3, 4syl2an 598 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
6 domentr 8558 . . . 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 8776 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
9 finnum 9368 . . . . 5 ((𝐴𝐵) ∈ Fin → (𝐴𝐵) ∈ dom card)
108, 9syl 17 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ dom card)
11 ficardom 9381 . . . . . 6 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
12 ficardom 9381 . . . . . 6 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
13 nnacl 8227 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
1411, 12, 13syl2an 598 . . . . 5 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ∈ ω)
15 nnon 7573 . . . . 5 (((card‘𝐴) +o (card‘𝐵)) ∈ ω → ((card‘𝐴) +o (card‘𝐵)) ∈ On)
16 onenon 9369 . . . . 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 9397 . . . 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 9383 . . 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 3955 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 1538   ∈ wcel 2111   ∪ cun 3879   ⊆ wss 3881   class class class wbr 5031  dom cdm 5520  Oncon0 6162  ‘cfv 6327  (class class class)co 7140  ωcom 7567   +o coa 8089   ≈ cen 8496   ≼ cdom 8497  Fincfn 8499   ⊔ cdju 9318  cardccrd 9355 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-dju 9321  df-card 9359 This theorem is referenced by: (None)
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