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 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)))

StepHypRef Expression
1 ficardom 9378 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
2 ficardom 9378 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
3 nnadju 9612 . . . . 5 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
4 df-dju 9318 . . . . . . 7 ((card‘𝐴) ⊔ (card‘𝐵)) = (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵)))
5 snfi 8581 . . . . . . . . 9 {∅} ∈ Fin
6 nnfi 8700 . . . . . . . . 9 ((card‘𝐴) ∈ ω → (card‘𝐴) ∈ Fin)
7 xpfi 8777 . . . . . . . . 9 (({∅} ∈ Fin ∧ (card‘𝐴) ∈ Fin) → ({∅} × (card‘𝐴)) ∈ Fin)
85, 6, 7sylancr 590 . . . . . . . 8 ((card‘𝐴) ∈ ω → ({∅} × (card‘𝐴)) ∈ Fin)
9 snfi 8581 . . . . . . . . 9 {1o} ∈ Fin
10 nnfi 8700 . . . . . . . . 9 ((card‘𝐵) ∈ ω → (card‘𝐵) ∈ Fin)
11 xpfi 8777 . . . . . . . . 9 (({1o} ∈ Fin ∧ (card‘𝐵) ∈ Fin) → ({1o} × (card‘𝐵)) ∈ Fin)
129, 10, 11sylancr 590 . . . . . . . 8 ((card‘𝐵) ∈ ω → ({1o} × (card‘𝐵)) ∈ Fin)
13 unfi 8773 . . . . . . . 8 ((({∅} × (card‘𝐴)) ∈ Fin ∧ ({1o} × (card‘𝐵)) ∈ Fin) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
148, 12, 13syl2an 598 . . . . . . 7 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
154, 14eqeltrid 2897 . . . . . 6 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin)
16 ficardid 9379 . . . . . 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 5057 . . . 4 (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
191, 2, 18syl2an 598 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
20 ficardid 9379 . . . 4 (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
21 ficardid 9379 . . . 4 (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
22 djuen 9584 . . . 4 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
2320, 21, 22syl2an 598 . . 3 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵))
24 entr 8548 . . 3 ((((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)) ∧ ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴𝐵)) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
2519, 23, 24syl2anc 587 . 2 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴𝐵))
2625ensymd 8547 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112   ∪ cun 3882  ∅c0 4246  {csn 4528   class class class wbr 5033   × cxp 5521  ‘cfv 6328  (class class class)co 7139  ωcom 7564  1oc1o 8082   +o coa 8086   ≈ cen 8493  Fincfn 8496   ⊔ cdju 9315  cardccrd 9352 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356 This theorem is referenced by:  ficardun  9615  ficardun2  9617
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