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Theorem disjssun 3608
 Description: Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjssun ((AB) = → (A (BC) ↔ A C))

Proof of Theorem disjssun
StepHypRef Expression
1 indi 3501 . . . . 5 (A ∩ (BC)) = ((AB) ∪ (AC))
21equncomi 3410 . . . 4 (A ∩ (BC)) = ((AC) ∪ (AB))
3 uneq2 3412 . . . . 5 ((AB) = → ((AC) ∪ (AB)) = ((AC) ∪ ))
4 un0 3575 . . . . 5 ((AC) ∪ ) = (AC)
53, 4syl6eq 2401 . . . 4 ((AB) = → ((AC) ∪ (AB)) = (AC))
62, 5syl5eq 2397 . . 3 ((AB) = → (A ∩ (BC)) = (AC))
76eqeq1d 2361 . 2 ((AB) = → ((A ∩ (BC)) = A ↔ (AC) = A))
8 df-ss 3259 . 2 (A (BC) ↔ (A ∩ (BC)) = A)
9 df-ss 3259 . 2 (A C ↔ (AC) = A)
107, 8, 93bitr4g 279 1 ((AB) = → (A (BC) ↔ A C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by: (None)
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