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Theorem unfiin 6563
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 496 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 ∈ Fin)
2 simpr 108 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
3 inss1 3204 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
43a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐴)
5 undiffi 6562 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
61, 2, 4, 5syl3anc 1170 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
7 simplr 497 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8 inss2 3205 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
98a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐵)
10 undiffi 6562 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
117, 2, 9, 10syl3anc 1170 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
126, 11uneq12d 3139 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵)))))
13 unundi 3145 . . . 4 ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
1412, 13syl6eqr 2133 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))))
15 diffifi 6540 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
161, 2, 4, 15syl3anc 1170 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
17 diffifi 6540 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
187, 2, 9, 17syl3anc 1170 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
19 incom 3176 . . . . . . . . . 10 (𝐵𝐴) = (𝐴𝐵)
2019difeq2i 3099 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵 ∖ (𝐴𝐵))
21 difin 3219 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
2220, 21eqtr3i 2105 . . . . . . . 8 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
2322ineq2i 3182 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴))
24 difss 3110 . . . . . . . 8 (𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴
25 disjdif 3337 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
26 ssdisj 3321 . . . . . . . 8 (((𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴 ∧ (𝐴 ∩ (𝐵𝐴)) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅)
2724, 25, 26mp2an 417 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅
2823, 27eqtri 2103 . . . . . 6 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅
2928a1i 9 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅)
30 unfidisj 6559 . . . . 5 (((𝐴 ∖ (𝐴𝐵)) ∈ Fin ∧ (𝐵 ∖ (𝐴𝐵)) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
3116, 18, 29, 30syl3anc 1170 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
32 difundir 3235 . . . . . . 7 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
3332ineq2i 3182 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))))
34 disjdif 3337 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ∅
3533, 34eqtr3i 2105 . . . . 5 ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅
3635a1i 9 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅)
37 unfidisj 6559 . . . 4 (((𝐴𝐵) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin ∧ ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
382, 31, 36, 37syl3anc 1170 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
3914, 38eqeltrd 2159 . 2 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
40393impa 1134 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  cdif 2981  cun 2982  cin 2983  wss 2984  c0 3269  Fincfn 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1o 6113  df-er 6222  df-en 6388  df-fin 6390
This theorem is referenced by: (None)
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