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Theorem ss2iun 3984
 Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ss2iun (x A B Cx A B x A C)

Proof of Theorem ss2iun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . . 5 (B C → (y By C))
21ralimi 2689 . . . 4 (x A B Cx A (y By C))
3 rexim 2718 . . . 4 (x A (y By C) → (x A y Bx A y C))
42, 3syl 15 . . 3 (x A B C → (x A y Bx A y C))
5 eliun 3973 . . 3 (y x A Bx A y B)
6 eliun 3973 . . 3 (y x A Cx A y C)
74, 5, 63imtr4g 261 . 2 (x A B C → (y x A By x A C))
87ssrdv 3278 1 (x A B Cx A B x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257  ∪ciun 3969 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-iun 3971 This theorem is referenced by:  iuneq2  3985
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