Theorem unfiin 7161

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 3429	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	3	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐴)
5		undiffi 7160	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))))
6	1, 2, 4, 5	syl3anc 1274	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))))
7		simplr 529	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8		inss2 3430	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9	8	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐵)
10		undiffi 7160	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
11	7, 2, 9, 10	syl3anc 1274	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
12	6, 11	uneq12d 3364	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
13		unundi 3370	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
14	12, 13	eqtr4di 2282	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
15		diffifi 7126	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
16	1, 2, 4, 15	syl3anc 1274	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
17		diffifi 7126	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
18	7, 2, 9, 17	syl3anc 1274	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
19		incom 3401	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
20	19	difeq2i 3324	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ (𝐴 ∩ 𝐵))
21		difin 3446	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴)
22	20, 21	eqtr3i 2254	. . . . . . . 8 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴)
23	22	ineq2i 3407	. . . . . . 7 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴))
24		difss 3335	. . . . . . . 8 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) ⊆ 𝐴
25		disjdif 3569	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
26		ssdisj 3553	. . . . . . . 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 2252	. . . . . 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 7157	. . . . 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 1274	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) ∈ Fin)
32		difundir 3462	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))
33	32	ineq2i 3407	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
34		disjdif 3569	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ∅
35	33, 34	eqtr3i 2254	. . . . 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 7157	. . . 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 1274	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) ∈ Fin)
39	14, 38	eqeltrd 2308	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
40	39	3impa 1221	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202   ∖ cdif 3198   ∪ cun 3199   ∩ cin 3200   ⊆ wss 3201  ∅c0 3496  Fincfn 6952

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-er 6745  df-en 6953  df-fin 6955

This theorem is referenced by:  4sqlem11  13037