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Theorem ficardadju 10240
Description: The disjoint union of finite sets is equinumerous to the ordinal sum of the cardinalities of those sets. (Contributed by BTernaryTau, 3-Jul-2024.)
Assertion
Ref Expression
ficardadju ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardadju
StepHypRef Expression
1 ficardom 10001 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
2 ficardom 10001 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
3 nnadju 10238 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
4 df-dju 9941 . . . . . . 7 ((card‘𝐴) ⊔ (card‘𝐵)) = (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵)))
5 snfi 9083 . . . . . . . . 9 {∅} ∈ Fin
6 nnfi 9207 . . . . . . . . 9 ((card‘𝐴) ∈ ω → (card‘𝐴) ∈ Fin)
7 xpfi 9358 . . . . . . . . 9 (({∅} ∈ Fin ∧ (card‘𝐴) ∈ Fin) → ({∅} × (card‘𝐴)) ∈ Fin)
85, 6, 7sylancr 587 . . . . . . . 8 ((card‘𝐴) ∈ ω → ({∅} × (card‘𝐴)) ∈ Fin)
9 snfi 9083 . . . . . . . . 9 {1o} ∈ Fin
10 nnfi 9207 . . . . . . . . 9 ((card‘𝐵) ∈ ω → (card‘𝐵) ∈ Fin)
11 xpfi 9358 . . . . . . . . 9 (({1o} ∈ Fin ∧ (card‘𝐵) ∈ Fin) → ({1o} × (card‘𝐵)) ∈ Fin)
129, 10, 11sylancr 587 . . . . . . . 8 ((card‘𝐵) ∈ ω → ({1o} × (card‘𝐵)) ∈ Fin)
13 unfi 9211 . . . . . . . 8 ((({∅} × (card‘𝐴)) ∈ Fin ∧ ({1o} × (card‘𝐵)) ∈ Fin) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
148, 12, 13syl2an 596 . . . . . . 7 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
154, 14eqeltrid 2845 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin)
16 ficardid 10002 . . . . . 6 (((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin → (card‘((card‘𝐴) ⊔ (card‘𝐵))) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
1715, 16syl 17 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
183, 17eqbrtrrd 5167 . . . 4 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
191, 2, 18syl2an 596 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
20 ficardid 10002 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
21 ficardid 10002 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
22 djuen 10210 . . . 4 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
2320, 21, 22syl2an 596 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
24 entr 9046 . . 3 ((((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)) ∧ ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵)) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
2519, 23, 24syl2anc 584 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
2625ensymd 9045 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  cun 3949  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499   +o coa 8503  cen 8982  Fincfn 8985  cdju 9938  cardccrd 9975
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979
This theorem is referenced by:  ficardun  10241  ficardun2  10242
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