Theorem inex2g 5291

Description: Sufficient condition for an intersection to be a set. Commuted form of inex1g 5290. (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 5290	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
3	1, 2	eqeltrid 2869	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3457   ∩ cin 3905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-in 3913

This theorem is used by:  ssexg  5292  dfac8b  10031  ac10ct  10034  satefvfmla1  35956  inex3  39047  inxpex  39048  dfcnvrefrels2  39317  dfcnvrefrels3  39318  iunrelexp0  44488