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Theorem iinss1 3981
 Description: Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
Assertion
Ref Expression
iinss1 (A Bx B C x A C)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   C(x)

Proof of Theorem iinss1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssralv 3330 . . 3 (A B → (x B y Cx A y C))
2 vex 2862 . . . 4 y V
3 eliin 3974 . . . 4 (y V → (y x B Cx B y C))
42, 3ax-mp 5 . . 3 (y x B Cx B y C)
5 eliin 3974 . . . 4 (y V → (y x A Cx A y C))
62, 5ax-mp 5 . . 3 (y x A Cx A y C)
71, 4, 63imtr4g 261 . 2 (A B → (y x B Cy x A C))
87ssrdv 3278 1 (A Bx B C x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ⊆ wss 3257  ∩ciin 3970 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iin 3972 This theorem is referenced by: (None)
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