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Theorem ssiun2 4009
 Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun2 (x AB x A B)

Proof of Theorem ssiun2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 rspe 2675 . . . 4 ((x A y B) → x A y B)
21ex 423 . . 3 (x A → (y Bx A y B))
3 eliun 3973 . . 3 (y x A Bx A y B)
42, 3syl6ibr 218 . 2 (x A → (y By x A B))
54ssrdv 3278 1 (x AB x A B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∃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:  ssiun2s  4010
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