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Theorem intminss 3952
 Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
Hypothesis
Ref Expression
intminss.1 (x = A → (φψ))
Assertion
Ref Expression
intminss ((A B ψ) → {x B φ} A)
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem intminss
StepHypRef Expression
1 intminss.1 . . 3 (x = A → (φψ))
21elrab 2994 . 2 (A {x B φ} ↔ (A B ψ))
3 intss1 3941 . 2 (A {x B φ} → {x B φ} A)
42, 3sylbir 204 1 ((A B ψ) → {x B φ} A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2618   ⊆ wss 3257  ∩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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by: (None)
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