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Theorem unfiin 6743
 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 ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)

Proof of Theorem unfiin
StepHypRef Expression
1 simpll 499 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 ∈ Fin)
2 simpr 109 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
3 inss1 3243 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
43a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐴)
5 undiffi 6742 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
61, 2, 4, 5syl3anc 1184 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
7 simplr 500 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8 inss2 3244 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
98a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐵)
10 undiffi 6742 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
117, 2, 9, 10syl3anc 1184 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
126, 11uneq12d 3178 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵)))))
13 unundi 3184 . . . 4 ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
1412, 13syl6eqr 2150 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))))
15 diffifi 6717 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
161, 2, 4, 15syl3anc 1184 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
17 diffifi 6717 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
187, 2, 9, 17syl3anc 1184 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
19 incom 3215 . . . . . . . . . 10 (𝐵𝐴) = (𝐴𝐵)
2019difeq2i 3138 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵 ∖ (𝐴𝐵))
21 difin 3260 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
2220, 21eqtr3i 2122 . . . . . . . 8 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
2322ineq2i 3221 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴))
24 difss 3149 . . . . . . . 8 (𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴
25 disjdif 3382 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
26 ssdisj 3366 . . . . . . . 8 (((𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴 ∧ (𝐴 ∩ (𝐵𝐴)) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅)
2724, 25, 26mp2an 420 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅
2823, 27eqtri 2120 . . . . . 6 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅
2928a1i 9 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅)
30 unfidisj 6739 . . . . 5 (((𝐴 ∖ (𝐴𝐵)) ∈ Fin ∧ (𝐵 ∖ (𝐴𝐵)) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
3116, 18, 29, 30syl3anc 1184 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
32 difundir 3276 . . . . . . 7 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
3332ineq2i 3221 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))))
34 disjdif 3382 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ∅
3533, 34eqtr3i 2122 . . . . 5 ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅
3635a1i 9 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅)
37 unfidisj 6739 . . . 4 (((𝐴𝐵) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin ∧ ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
382, 31, 36, 37syl3anc 1184 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
3914, 38eqeltrd 2176 . 2 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
40393impa 1144 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 930   = wceq 1299   ∈ wcel 1448   ∖ cdif 3018   ∪ cun 3019   ∩ cin 3020   ⊆ wss 3021  ∅c0 3310  Fincfn 6564 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440 This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1o 6243  df-er 6359  df-en 6565  df-fin 6567 This theorem is referenced by: (None)
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