Theorem intpr 4987

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 4986. (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 4986	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
4	1, 2, 3	mp2an 692	1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∩ cin 3962  {cpr 4633  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-in 3970  df-sn 4632  df-pr 4634  df-int 4952

This theorem is referenced by:  uniintsn  4990  op1stb  5482  fiint  9364  fiintOLD  9365  shincli  31391  chincli  31489