Theorem intmin3 4973

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 3661	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	mpbiri 258	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝑥 ∣ 𝜑})
5		intss1 4960	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)
6	4, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cab 2703   ⊆ wss 3943  ∩ cint 4943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-int 4944

This theorem is referenced by:  intabs  5335  intidOLD  5451