Theorem intpr 4951

Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) Prove from intprg 4950. (Revised by BJ, 1-Sep-2024.)

Hypotheses

Ref	Expression
intpr.1	⊢ 𝐴 ∈ V
intpr.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
intpr	⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)

Proof of Theorem intpr

Step	Hyp	Ref	Expression
1		intpr.1	. 2 ⊢ 𝐴 ∈ V
2		intpr.2	. 2 ⊢ 𝐵 ∈ V
3		intprg 4950	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
4	1, 2, 3	mp2an 704	1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  {cpr 4596  ∩ cint 4916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-in 3920  df-sn 4595  df-pr 4597  df-int 4917

This theorem is referenced by:  uniintsn  4954  op1stb  5454  fiint  9286  shincli  31655  chincli  31753