Theorem intminss 3953

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 → (φ ↔ ψ))

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

Step	Hyp	Ref	Expression
1		intminss.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
2	1	elrab 2995	. 2 ⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ))
3		intss1 3942	. 2 ⊢ (A ∈ {x ∈ B ∣ φ} → ∩{x ∈ B ∣ φ} ⊆ A)
4	2, 3	sylbir 204	1 ⊢ ((A ∈ B ∧ ψ) → ∩{x ∈ B ∣ φ} ⊆ A)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2619   ⊆ wss 3258  ∩cint 3927

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 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by: (None)