Theorem unfiin 7223

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 531	. . . . . 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 3451	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	3	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐴)
5		undiffi 7222	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))))
6	1, 2, 4, 5	syl3anc 1278	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))))
7		simplr 533	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8		inss2 3452	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9	8	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∩ 𝐵) ⊆ 𝐵)
10		undiffi 7222	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
11	7, 2, 9, 10	syl3anc 1278	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → 𝐵 = ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
12	6, 11	uneq12d 3384	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
13		unundi 3390	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) = (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ (𝐴 ∩ 𝐵))) ∪ ((𝐴 ∩ 𝐵) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
14	12, 13	eqtr4di 2289	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) = ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))))
15		diffifi 7188	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
16	1, 2, 4, 15	syl3anc 1278	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
17		diffifi 7188	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
18	7, 2, 9, 17	syl3anc 1278	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐵 ∖ (𝐴 ∩ 𝐵)) ∈ Fin)
19		incom 3421	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
20	19	difeq2i 3344	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ (𝐴 ∩ 𝐵))
21		difin 3468	. . . . . . . . 9 ⊢ (𝐵 ∖ (𝐵 ∩ 𝐴)) = (𝐵 ∖ 𝐴)
22	20, 21	eqtr3i 2261	. . . . . . . 8 ⊢ (𝐵 ∖ (𝐴 ∩ 𝐵)) = (𝐵 ∖ 𝐴)
23	22	ineq2i 3429	. . . . . . 7 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴))
24		difss 3355	. . . . . . . 8 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) ⊆ 𝐴
25		disjdif 3596	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
26		ssdisj 3580	. . . . . . . 8 ⊢ (((𝐴 ∖ (𝐴 ∩ 𝐵)) ⊆ 𝐴 ∧ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴)) = ∅)
27	24, 25, 26	mp2an 430	. . . . . . 7 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∩ (𝐵 ∖ 𝐴)) = ∅
28	23, 27	eqtri 2259	. . . . . 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 7219	. . . . 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 1278	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))) ∈ Fin)
32		difundir 3484	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) = ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))
33	32	ineq2i 3429	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵))))
34		disjdif 3596	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∩ ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵))) = ∅
35	33, 34	eqtr3i 2261	. . . . 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 7219	. . . 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 1278	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → ((𝐴 ∩ 𝐵) ∪ ((𝐴 ∖ (𝐴 ∩ 𝐵)) ∪ (𝐵 ∖ (𝐴 ∩ 𝐵)))) ∈ Fin)
39	14, 38	eqeltrd 2315	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
40	39	3impa 1225	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  Fincfn 7012

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015

This theorem is referenced by:  4sqlem11  13158