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Theorem ssuni 3913
 Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ssuni ((A B B C) → A C)

Proof of Theorem ssuni
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . . . . 7 (x = B → (y xy B))
21imbi1d 308 . . . . . 6 (x = B → ((y xy C) ↔ (y By C)))
3 elunii 3896 . . . . . . 7 ((y x x C) → y C)
43expcom 424 . . . . . 6 (x C → (y xy C))
52, 4vtoclga 2920 . . . . 5 (B C → (y By C))
65imim2d 48 . . . 4 (B C → ((y Ay B) → (y Ay C)))
76alimdv 1621 . . 3 (B C → (y(y Ay B) → y(y Ay C)))
8 dfss2 3262 . . 3 (A By(y Ay B))
9 dfss2 3262 . . 3 (A Cy(y Ay C))
107, 8, 93imtr4g 261 . 2 (B C → (A BA C))
1110impcom 419 1 ((A B B C) → A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ∪cuni 3891 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-ss 3259  df-uni 3892 This theorem is referenced by:  elssuni  3919  uniss2  3922
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