Theorem unfiin 6984

Description: The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)

Assertion

Ref	Expression
unfiin	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Proof of Theorem unfiin

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐴 ∈ Fin)
2		simpr 110	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ∈ Fin)
3		inss1 3380	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	3	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐴)
5		undiffi 6983	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))))
6	1, 2, 4, 5	syl3anc 1249	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))))
7		simplr 528	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8		inss2 3381	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9	8	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐵)
10		undiffi 6983	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
11	7, 2, 9, 10	syl3anc 1249	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
12	6, 11	uneq12d 3315	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
13		unundi 3321	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
14	12, 13	eqtr4di 2244	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
15		diffifi 6952	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
16	1, 2, 4, 15	syl3anc 1249	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
17		diffifi 6952	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
18	7, 2, 9, 17	syl3anc 1249	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
19		incom 3352	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
20	19	difeq2i 3275	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ (𝐴 ∩ 𝐵))
21		difin 3397	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴)
22	20, 21	eqtr3i 2216	. . . . . . . 8 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴)
23	22	ineq2i 3358	. . . . . . 7 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴))
24		difss 3286	. . . . . . . 8 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) ⊆ 𝐴
25		disjdif 3520	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
26		ssdisj 3504	. . . . . . . 8 ⊢ (((𝐴 ∖ (𝐴 ∩ 𝐵)) ⊆ 𝐴 ∧ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴)) = ∅)
27	24, 25, 26	mp2an 426	. . . . . . 7 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴)) = ∅
28	23, 27	eqtri 2214	. . . . . 6 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ∅
29	28	a1i 9	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ∅)
30		unfidisj 6980	. . . . 5 ⊢ (((𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin ∧ (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin ∧ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ∅) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) ∈ Fin)
31	16, 18, 29, 30	syl3anc 1249	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) ∈ Fin)
32		difundir 3413	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))
33	32	ineq2i 3358	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
34		disjdif 3520	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ∅
35	33, 34	eqtr3i 2216	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) = ∅
36	35	a1i 9	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) = ∅)
37		unfidisj 6980	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ Fin ∧ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) ∈ Fin ∧ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) = ∅) → ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) ∈ Fin)
38	2, 31, 36, 37	syl3anc 1249	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) ∈ Fin)
39	14, 38	eqeltrd 2270	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
40	39	3impa 1196	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2164   ∖ cdif 3151   ∪ cun 3152   ∩ cin 3153   ⊆ wss 3154  ∅c0 3447  Fincfn 6796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6471  df-er 6589  df-en 6797  df-fin 6799

This theorem is referenced by:  4sqlem11  12542