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Theorem intssuni 3948
 Description: The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssuni (AA A)

Proof of Theorem intssuni
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r19.2z 3639 . . . 4 ((A y A x y) → y A x y)
21ex 423 . . 3 (A → (y A x yy A x y))
3 vex 2862 . . . 4 x V
43elint2 3933 . . 3 (x Ay A x y)
5 eluni2 3895 . . 3 (x Ay A x y)
62, 4, 53imtr4g 261 . 2 (A → (x Ax A))
76ssrdv 3278 1 (AA A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257  ∅c0 3550  ∪cuni 3891  ∩cint 3926 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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-uni 3892  df-int 3927 This theorem is referenced by:  unissint  3950  intssuni2  3951
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