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Theorem unfiin 6782
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 503 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 ∈ Fin)
2 simpr 109 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
3 inss1 3266 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
43a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐴)
5 undiffi 6781 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
61, 2, 4, 5syl3anc 1201 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
7 simplr 504 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8 inss2 3267 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
98a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐵)
10 undiffi 6781 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
117, 2, 9, 10syl3anc 1201 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
126, 11uneq12d 3201 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵)))))
13 unundi 3207 . . . 4 ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
1412, 13syl6eqr 2168 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))))
15 diffifi 6756 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
161, 2, 4, 15syl3anc 1201 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
17 diffifi 6756 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
187, 2, 9, 17syl3anc 1201 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
19 incom 3238 . . . . . . . . . 10 (𝐵𝐴) = (𝐴𝐵)
2019difeq2i 3161 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵 ∖ (𝐴𝐵))
21 difin 3283 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
2220, 21eqtr3i 2140 . . . . . . . 8 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
2322ineq2i 3244 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴))
24 difss 3172 . . . . . . . 8 (𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴
25 disjdif 3405 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
26 ssdisj 3389 . . . . . . . 8 (((𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴 ∧ (𝐴 ∩ (𝐵𝐴)) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅)
2724, 25, 26mp2an 422 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅
2823, 27eqtri 2138 . . . . . 6 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅
2928a1i 9 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅)
30 unfidisj 6778 . . . . 5 (((𝐴 ∖ (𝐴𝐵)) ∈ Fin ∧ (𝐵 ∖ (𝐴𝐵)) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
3116, 18, 29, 30syl3anc 1201 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
32 difundir 3299 . . . . . . 7 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
3332ineq2i 3244 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))))
34 disjdif 3405 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ∅
3533, 34eqtr3i 2140 . . . . 5 ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅
3635a1i 9 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅)
37 unfidisj 6778 . . . 4 (((𝐴𝐵) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin ∧ ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
382, 31, 36, 37syl3anc 1201 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
3914, 38eqeltrd 2194 . 2 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
40393impa 1161 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wcel 1465  cdif 3038  cun 3039  cin 3040  wss 3041  c0 3333  Fincfn 6602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1o 6281  df-er 6397  df-en 6603  df-fin 6605
This theorem is referenced by: (None)
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