Theorem inex2g 5283

Description: Sufficient condition for an intersection to be a set. Commuted form of inex1g 5282. (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 4108	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inex1g 5282	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
3	1, 2	eqeltrid 2867	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3453   ∩ cin 4091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-in 4093

This theorem is used by:  ssexg  5284  dfac8b  10120  ac10ct  10123  satefvfmla1  36311  inex3  39410  inxpex  39411  dfcnvrefrels2  39680  dfcnvrefrels3  39681  iunrelexp0  44842