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Theorem ssiun 4008
 Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun (x A C BC x A B)
Distinct variable group:   x,C
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem ssiun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . . 5 (C B → (y Cy B))
21reximi 2721 . . . 4 (x A C Bx A (y Cy B))
3 r19.37av 2761 . . . 4 (x A (y Cy B) → (y Cx A y B))
42, 3syl 15 . . 3 (x A C B → (y Cx A y B))
5 eliun 3973 . . 3 (y x A Bx A y B)
64, 5syl6ibr 218 . 2 (x A C B → (y Cy x A B))
76ssrdv 3278 1 (x A C BC 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-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:  iunss2  4011  iunpwss  4055
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