Theorem intpr 4985

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 4984. (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 4984	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
4	1, 2, 3	mp2an 690	1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3946  {cpr 4629  ∩ cint 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3952  df-in 3954  df-sn 4628  df-pr 4630  df-int 4950

This theorem is referenced by:  intprgOLD  4987  uniintsn  4990  op1stb  5470  fiint  9320  shincli  30602  chincli  30700