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Theorem unfiin 6814
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 518 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 ∈ Fin)
2 simpr 109 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
3 inss1 3296 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
43a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐴)
5 undiffi 6813 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
61, 2, 4, 5syl3anc 1216 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐴 = ((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))))
7 simplr 519 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 ∈ Fin)
8 inss2 3297 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
98a1i 9 . . . . . 6 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ⊆ 𝐵)
10 undiffi 6813 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
117, 2, 9, 10syl3anc 1216 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → 𝐵 = ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
126, 11uneq12d 3231 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵)))))
13 unundi 3237 . . . 4 ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = (((𝐴𝐵) ∪ (𝐴 ∖ (𝐴𝐵))) ∪ ((𝐴𝐵) ∪ (𝐵 ∖ (𝐴𝐵))))
1412, 13eqtr4di 2190 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) = ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))))
15 diffifi 6788 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
161, 2, 4, 15syl3anc 1216 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴 ∖ (𝐴𝐵)) ∈ Fin)
17 diffifi 6788 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin ∧ (𝐴𝐵) ⊆ 𝐵) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
187, 2, 9, 17syl3anc 1216 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐵 ∖ (𝐴𝐵)) ∈ Fin)
19 incom 3268 . . . . . . . . . 10 (𝐵𝐴) = (𝐴𝐵)
2019difeq2i 3191 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵 ∖ (𝐴𝐵))
21 difin 3313 . . . . . . . . 9 (𝐵 ∖ (𝐵𝐴)) = (𝐵𝐴)
2220, 21eqtr3i 2162 . . . . . . . 8 (𝐵 ∖ (𝐴𝐵)) = (𝐵𝐴)
2322ineq2i 3274 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴))
24 difss 3202 . . . . . . . 8 (𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴
25 disjdif 3435 . . . . . . . 8 (𝐴 ∩ (𝐵𝐴)) = ∅
26 ssdisj 3419 . . . . . . . 8 (((𝐴 ∖ (𝐴𝐵)) ⊆ 𝐴 ∧ (𝐴 ∩ (𝐵𝐴)) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅)
2724, 25, 26mp2an 422 . . . . . . 7 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵𝐴)) = ∅
2823, 27eqtri 2160 . . . . . 6 ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅
2928a1i 9 . . . . 5 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅)
30 unfidisj 6810 . . . . 5 (((𝐴 ∖ (𝐴𝐵)) ∈ Fin ∧ (𝐵 ∖ (𝐴𝐵)) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∩ (𝐵 ∖ (𝐴𝐵))) = ∅) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
3116, 18, 29, 30syl3anc 1216 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin)
32 difundir 3329 . . . . . . 7 ((𝐴𝐵) ∖ (𝐴𝐵)) = ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))
3332ineq2i 3274 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))))
34 disjdif 3435 . . . . . 6 ((𝐴𝐵) ∩ ((𝐴𝐵) ∖ (𝐴𝐵))) = ∅
3533, 34eqtr3i 2162 . . . . 5 ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅
3635a1i 9 . . . 4 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅)
37 unfidisj 6810 . . . 4 (((𝐴𝐵) ∈ Fin ∧ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵))) ∈ Fin ∧ ((𝐴𝐵) ∩ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) = ∅) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
382, 31, 36, 37syl3anc 1216 . . 3 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → ((𝐴𝐵) ∪ ((𝐴 ∖ (𝐴𝐵)) ∪ (𝐵 ∖ (𝐴𝐵)))) ∈ Fin)
3914, 38eqeltrd 2216 . 2 (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
40393impa 1176 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wcel 1480  cdif 3068  cun 3069  cin 3070  wss 3071  c0 3363  Fincfn 6634
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313  df-er 6429  df-en 6635  df-fin 6637
This theorem is referenced by: (None)
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