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Theorem uniss2 3922
 Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4011 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
uniss2 (x A y B x yA B)
Distinct variable groups:   x,A   x,y,B
Allowed substitution hint:   A(y)

Proof of Theorem uniss2
StepHypRef Expression
1 ssuni 3913 . . . . 5 ((x y y B) → x B)
21expcom 424 . . . 4 (y B → (x yx B))
32rexlimiv 2732 . . 3 (y B x yx B)
43ralimi 2689 . 2 (x A y B x yx A x B)
5 unissb 3921 . 2 (A Bx A x B)
64, 5sylibr 203 1 (x A y B x yA B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ⊆ 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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-uni 3892 This theorem is referenced by:  unidif  3923
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