Theorem inex2g 5311

Description: Sufficient condition for an intersection to be a set. Commuted form of inex1g 5310. (Contributed by Peter Mazsa, 19-Dec-2018.)

Assertion

Ref	Expression
inex2g	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)

Proof of Theorem inex2g

Step	Hyp	Ref	Expression
1		incom 4194	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inex1g 5310	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
3	1, 2	eqeltrid 2829	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3466   ∩ cin 3940

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 2695  ax-sep 5290

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-in 3948

This theorem is referenced by:  satefvfmla1  34934  inex3  37711  inxpex  37712  dfcnvrefrels2  37902  dfcnvrefrels3  37903  iunrelexp0  43003