Theorem inex3 37207

Description: Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.)

Assertion

Ref	Expression
inex3	⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)

Proof of Theorem inex3

Step	Hyp	Ref	Expression
1		inex1g 5320	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
2		inex2g 5321	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V)
3	1, 2	jaoi 856	1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∨ wo 846   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956

This theorem is referenced by: (None)