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Theorem unfiin 6954
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 527 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 ∈ Fin)
2 simpr 110 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
3 inss1 3370 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
43a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐴)
5 undiffi 6953 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
61, 2, 4, 5syl3anc 1249 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
7 simplr 528 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8 inss2 3371 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
98a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐵)
10 undiffi 6953 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
117, 2, 9, 10syl3anc 1249 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
126, 11uneq12d 3305 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵)))))
13 unundi 3311 . . . 4 ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
1412, 13eqtr4di 2240 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))))
15 diffifi 6922 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
161, 2, 4, 15syl3anc 1249 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
17 diffifi 6922 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
187, 2, 9, 17syl3anc 1249 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
19 incom 3342 . . . . . . . . . 10 (𝐵𝐴) = (𝐴𝐵)
2019difeq2i 3265 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵 ∖ (𝐴𝐵))
21 difin 3387 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
2220, 21eqtr3i 2212 . . . . . . . 8 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
2322ineq2i 3348 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴))
24 difss 3276 . . . . . . . 8 (𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴
25 disjdif 3510 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
26 ssdisj 3494 . . . . . . . 8 (((𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴 ∧ (𝐴 ∩ (𝐵𝐴)) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅)
2724, 25, 26mp2an 426 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅
2823, 27eqtri 2210 . . . . . 6 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅
2928a1i 9 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅)
30 unfidisj 6950 . . . . 5 (((𝐴 ∖ (𝐴𝐵)) ∈ Fin ∧ (𝐵 ∖ (𝐴𝐵)) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
3116, 18, 29, 30syl3anc 1249 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
32 difundir 3403 . . . . . . 7 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
3332ineq2i 3348 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))))
34 disjdif 3510 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ∅
3533, 34eqtr3i 2212 . . . . 5 ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅
3635a1i 9 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅)
37 unfidisj 6950 . . . 4 (((𝐴𝐵) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin ∧ ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
382, 31, 36, 37syl3anc 1249 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
3914, 38eqeltrd 2266 . 2 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
40393impa 1196 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  cdif 3141  cun 3142  cin 3143  wss 3144  c0 3437  Fincfn 6766
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1o 6441  df-er 6559  df-en 6767  df-fin 6769
This theorem is referenced by:  4sqlem11  12433
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