Theorem ficardadju 10091

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 9854	. . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
2		ficardom 9854	. . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
3		nnadju 10089	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (card‘((card‘𝐴) ⊔ (card‘𝐵))) = ((card‘𝐴) +o (card‘𝐵)))
4		df-dju 9794	. . . . . . 7 ⊢ ((card‘𝐴) ⊔ (card‘𝐵)) = (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵)))
5		snfi 8965	. . . . . . . . 9 ⊢ {∅} ∈ Fin
6		nnfi 9077	. . . . . . . . 9 ⊢ ((card‘𝐴) ∈ ω → (card‘𝐴) ∈ Fin)
7		xpfi 9204	. . . . . . . . 9 ⊢ (({∅} ∈ Fin ∧ (card‘𝐴) ∈ Fin) → ({∅} × (card‘𝐴)) ∈ Fin)
8	5, 6, 7	sylancr 587	. . . . . . . 8 ⊢ ((card‘𝐴) ∈ ω → ({∅} × (card‘𝐴)) ∈ Fin)
9		snfi 8965	. . . . . . . . 9 ⊢ {1o} ∈ Fin
10		nnfi 9077	. . . . . . . . 9 ⊢ ((card‘𝐵) ∈ ω → (card‘𝐵) ∈ Fin)
11		xpfi 9204	. . . . . . . . 9 ⊢ (({1o} ∈ Fin ∧ (card‘𝐵) ∈ Fin) → ({1o} × (card‘𝐵)) ∈ Fin)
12	9, 10, 11	sylancr 587	. . . . . . . 8 ⊢ ((card‘𝐵) ∈ ω → ({1o} × (card‘𝐵)) ∈ Fin)
13		unfi 9080	. . . . . . . 8 ⊢ ((({∅} × (card‘𝐴)) ∈ Fin ∧ ({1o} × (card‘𝐵)) ∈ Fin) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
14	8, 12, 13	syl2an 596	. . . . . . 7 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → (({∅} × (card‘𝐴)) ∪ ({1o} × (card‘𝐵))) ∈ Fin)
15	4, 14	eqeltrid 2835	. . . . . 6 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊔ (card‘𝐵)) ∈ Fin)
16		ficardid 9855	. . . . . 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 5113	. . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
19	1, 2, 18	syl2an 596	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ ((card‘𝐴) ⊔ (card‘𝐵)))
20		ficardid 9855	. . . 4 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
21		ficardid 9855	. . . 4 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
22		djuen 10061	. . . 4 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
23	20, 21, 22	syl2an 596	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊔ (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
24		entr 8928	. . 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 584	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) +o (card‘𝐵)) ≈ (𝐴 ⊔ 𝐵))
26	25	ensymd 8927	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111   ∪ cun 3895  ∅c0 4280  {csn 4573   class class class wbr 5089   × cxp 5612  ‘cfv 6481  (class class class)co 7346  ωcom 7796  1_oc1o 8378   +_o coa 8382   ≈ cen 8866  Fincfn 8869   ⊔ cdju 9791  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832

This theorem is referenced by:  ficardun  10092  ficardun2  10093