Theorem ficardadju 9644

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) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))

Proof of Theorem ficardadju

Step	Hyp	Ref	Expression
1		ficardom 9408	. . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
2		ficardom 9408	. . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
3		nnadju 9642	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
4		df-dju 9348	. . . . . . 7 ⊢ ((card‘𝐴) ⊔ (card‘𝐵)) = (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵)))
5		snfi 8607	. . . . . . . . 9 ⊢ {∅} ∈ Fin
6		nnfi 8726	. . . . . . . . 9 ⊢ ((card‘𝐴) ∈ ω → (card‘𝐴) ∈ Fin)
7		xpfi 8807	. . . . . . . . 9 ⊢ (({∅} ∈ Fin ∧ (card‘𝐴) ∈ Fin) → ({∅} × (card‘𝐴)) ∈ Fin)
8	5, 6, 7	sylancr 591	. . . . . . . 8 ⊢ ((card‘𝐴) ∈ ω → ({∅} × (card‘𝐴)) ∈ Fin)
9		snfi 8607	. . . . . . . . 9 ⊢ {1o} ∈ Fin
10		nnfi 8726	. . . . . . . . 9 ⊢ ((card‘𝐵) ∈ ω → (card‘𝐵) ∈ Fin)
11		xpfi 8807	. . . . . . . . 9 ⊢ (({1o} ∈ Fin ∧ (card‘𝐵) ∈ Fin) → ({1o} × (card‘𝐵)) ∈ Fin)
12	9, 10, 11	sylancr 591	. . . . . . . 8 ⊢ ((card‘𝐵) ∈ ω → ({1o} × (card‘𝐵)) ∈ Fin)
13		unfi 8803	. . . . . . . 8 ⊢ ((({∅} × (card‘𝐴)) ∈ Fin ∧ ({1o} × (card‘𝐵)) ∈ Fin) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
14	8, 12, 13	syl2an 599	. . . . . . 7 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
15	4, 14	eqeltrid 2855	. . . . . 6 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin)
16		ficardid 9409	. . . . . 6 ⊢ (((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin → (card‘((card‘𝐴) ⊔ (card‘𝐵))) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
17	15, 16	syl 17	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
18	3, 17	eqbrtrrd 5049	. . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
19	1, 2, 18	syl2an 599	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
20		ficardid 9409	. . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
21		ficardid 9409	. . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
22		djuen 9614	. . . 4 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
23	20, 21, 22	syl2an 599	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
24		entr 8572	. . 3 ⊢ ((((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)) ∧ ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵)) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
25	19, 23, 24	syl2anc 588	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
26	25	ensymd 8571	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2112   ∪ cun 3852  ∅c0 4221  {csn 4515   class class class wbr 5025   × cxp 5515  ‘cfv 6328  (class class class)co 7143  ωcom 7572  1_oc1o 8098   +_o coa 8102   ≈ cen 8517  Fincfn 8520   ⊔ cdju 9345  cardccrd 9382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  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 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-dju 9348  df-card 9386

This theorem is referenced by:  ficardun  9645  ficardun2  9647