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Theorem iunss2 4011
 Description: A subclass condition on the members of two indexed classes C(x) and D(y) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3922. (Contributed by NM, 9-Dec-2004.)
Assertion
Ref Expression
iunss2 (x A y B C Dx A C y B D)
Distinct variable groups:   x,y   x,B   y,C   x,D
Allowed substitution hints:   A(x,y)   B(y)   C(x)   D(y)

Proof of Theorem iunss2
StepHypRef Expression
1 ssiun 4008 . . 3 (y B C DC y B D)
21ralimi 2689 . 2 (x A y B C Dx A C y B D)
3 iunss 4007 . 2 (x A C y B Dx A C y B D)
42, 3sylibr 203 1 (x A y B C Dx A C y B D)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257  ∪ciun 3969 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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971 This theorem is referenced by:  iunxdif2  4014
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