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Theorem uniss 3912
 Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniss (A BA B)

Proof of Theorem uniss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . . 5 (A B → (y Ay B))
21anim2d 548 . . . 4 (A B → ((x y y A) → (x y y B)))
32eximdv 1622 . . 3 (A B → (y(x y y A) → y(x y y B)))
4 eluni 3894 . . 3 (x Ay(x y y A))
5 eluni 3894 . . 3 (x By(x y y B))
63, 4, 53imtr4g 261 . 2 (A B → (x Ax B))
76ssrdv 3278 1 (A BA B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ⊆ wss 3257  ∪cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  unissi  3914  unissd  3915  unidif  3923  intssuni2  3951  uniintsn  3963  sspw1  4335
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