Theorem unfiin 7084

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 3424	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	3	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐴)
5		undiffi 7083	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))))
6	1, 2, 4, 5	syl3anc 1271	. . . . 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 3425	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9	8	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐵)
10		undiffi 7083	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
11	7, 2, 9, 10	syl3anc 1271	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
12	6, 11	uneq12d 3359	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
13		unundi 3365	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
14	12, 13	eqtr4di 2280	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
15		diffifi 7052	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
16	1, 2, 4, 15	syl3anc 1271	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
17		diffifi 7052	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
18	7, 2, 9, 17	syl3anc 1271	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
19		incom 3396	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
20	19	difeq2i 3319	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ (𝐴 ∩ 𝐵))
21		difin 3441	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴)
22	20, 21	eqtr3i 2252	. . . . . . . 8 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴)
23	22	ineq2i 3402	. . . . . . 7 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴))
24		difss 3330	. . . . . . . 8 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) ⊆ 𝐴
25		disjdif 3564	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
26		ssdisj 3548	. . . . . . . 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 2250	. . . . . 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 7080	. . . . 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 1271	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) ∈ Fin)
32		difundir 3457	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))
33	32	ineq2i 3402	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
34		disjdif 3564	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ∅
35	33, 34	eqtr3i 2252	. . . . 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 7080	. . . 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 1271	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) ∈ Fin)
39	14, 38	eqeltrd 2306	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
40	39	3impa 1218	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ∖ cdif 3194   ∪ cun 3195   ∩ cin 3196   ⊆ wss 3197  ∅c0 3491  Fincfn 6885

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-er 6678  df-en 6886  df-fin 6888

This theorem is referenced by:  4sqlem11  12919