Theorem intmin3 3954

Description: Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)

Hypotheses

Ref	Expression
intmin3.2	⊢ (x = A → (φ ↔ ψ))
intmin3.3	⊢ ψ

Assertion

Ref	Expression
intmin3	⊢ (A ∈ V → ∩{x ∣ φ} ⊆ A)

Distinct variable groups:   x,A   ψ,x

Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem intmin3

Step	Hyp	Ref	Expression
1		intmin3.3	. . 3 ⊢ ψ
2		intmin3.2	. . . 4 ⊢ (x = A → (φ ↔ ψ))
3	2	elabg 2986	. . 3 ⊢ (A ∈ V → (A ∈ {x ∣ φ} ↔ ψ))
4	1, 3	mpbiri 224	. 2 ⊢ (A ∈ V → A ∈ {x ∣ φ})
5		intss1 3941	. 2 ⊢ (A ∈ {x ∣ φ} → ∩{x ∣ φ} ⊆ A)
6	4, 5	syl 15	1 ⊢ (A ∈ V → ∩{x ∣ φ} ⊆ A)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927

This theorem is referenced by: (None)