Theorem inex2g 5289

Description: Sufficient condition for an intersection to be a set. Commuted form of inex1g 5288. (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 4162	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inex1g 5288	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
3	1, 2	eqeltrid 2867	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455   ∩ cin 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3912

This theorem is referenced by:  ssexg  5290  satefvfmla1  35917  inex3  39007  inxpex  39008  dfcnvrefrels2  39277  dfcnvrefrels3  39278  iunrelexp0  44448