Theorem intpr 4942

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 4941. (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 4941	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
4	1, 2, 3	mp2an 705	1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898  {cpr 4586  ∩ cint 4907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-in 3906  df-sn 4585  df-pr 4587  df-int 4908

This theorem is used by:  uniintsn  4945  op1stb  5447  fiint  9299  shincli  31846  chincli  31944